Existence of Integral m -Varifolds minimizing ∫|A | p and ∫|H | p , p>m , in Riemannian Manifolds
aa r X i v : . [ m a t h . DG ] J a n Existence of Integral m -Varifolds minimizing R | A | p and R | H | p , p > m , in Riemannian Manifolds Andrea MONDINO abstract . We prove existence of integral rectifiable m -dimensional varifolds minimizing functionals ofthe type R | H | p and R | A | p in a given Riemannian n -dimensional manifold ( N, g ), 2 ≤ m < n and p > m , under suitable assumptions on N (in the end of the paper we give many examples of suchambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas forvarifolds in R S involving R | H | p , to avoid degeneracy of the minimizer, and a sort of isoperimetricinequality to bound the mass in terms of the mentioned functionals. Key Words:
Curvature varifolds, direct methods in the calculus of variation, geometric measure theory,isoperimetric inequality, monotonicity formula
AMS subject classification:
Given an ambient Riemannian manifold (
N, g ) of dimension n ≥ m -dimensional submanifolds, 2 ≤ m ≤ n −
1, with null mean curvature vector, H = 0, or with null second fundamental form, A = 0, namelythe minimal (respectively, the totally geodesic) submanifolds of N (for more details about the existencesee Example 7.3, Example 7.4, Theorem 7.6, Theorem 7.7, Remark 7.8 and Remark 7.9).In more generality, it is interesting to study the minimization problems associated to integral func-tionals depending on the curvatures of the type(1) E pH,m ( M ) := Z M | H | p or E pA,m ( M ) := Z M | A | p , p ≥ M is a smooth immersed m -dimensional submanifold with mean curvature H and second funda-mental form A ; of course the integrals are computed with respect to the m -dimensional measure of N induced on M . A global minimizer, if it exists, of E pH,m (respectively of E pA,m ) can be seen as a generalizedminimal (respectively totally geodesic) m -dimensional submanifold in a natural integral sense.An important example of such functionals is given by the Willmore functional for surfaces E H, introduced by Willmore (see [33]) and studied in the euclidean space (see for instance the works of Simon[27], Kuwert and Sch¨atzle [12], Rivi`ere [30]) or in Riemannian manifolds (see, for example, [13], [21] and[22]).The general integral functionals (1) depending on the curvatures of immersed submanifolds have beenstudied, among others, by Allard [1], Anzellotti-Serapioni-Tamanini [3], Delladio [6], Hutchinson [9], [10],[11], Mantegazza [16] and Moser [24]. SISSA, Via Bonomea, 265, 34136 Trieste, Italy, E-mail address: [email protected], Tel: +39 040 3787-293
1n order to get the existence of a minimizer, the technique adopted in the present paper (as wellas in most of the aforementioned ones) is the so called direct method in the calculus of variations. Asusual, it is necessary to enlarge the space where the functional is defined and to work out a compactness-lowersemicontinuity theory in the enlarged domain.In the present work, the enlarged domain is made of generalized m -dimensional submanifolds of thefixed ambient Riemannian manifold ( N, g ): the integral rectifiable m -varifolds introduced by Almgren in[2] and by Allard in [1]. Using integration by parts formulas, Allard [1] and Hutchinson [9]-Mantegazza[16] defined a weak notion of mean curvature and of second fundamental form respectively (for moredetails about this part see Section 2). Moreover these objects have good compactness and lower semi-continuity properties with respect to the integral functionals above.The goal of this paper is to prove existence and partial regularity of an m -dimensional minimizer (inthe enlarged class of the rectifiable integral m -varifolds with weak mean curvature or with generalizedsecond fundamental form in the sense explained above) of functionals of the type (1). Actually we willconsider more general functionals modeled on this example, see Definition 2.2 for the expression of theconsidered integrand F .More precisely, given a compact subset N ⊂⊂ ¯ N of an n -dimensional Riemannian manifold ( ¯ N , g )(which, by Nash Embedding Theorem, can be assumed isometrically embedded in some R S ) we willdenote HV m ( N ) := { V integral rectifiable m -varifold of N with weak mean curvature H N relative to ¯ N } CV m ( N ) := { V integral rectifiable m -varifold of N with generalized second fundamental form A } ;for more details see Section 2; in any case, as written above, the non expert reader can think about theelements of HV m ( N ) (respectively of CV m ( N )) as generalized m -dimensional submanifolds with meancurvature H N (respectively with second fundamental form A ). Precisely, we consider the following twominimization problems(2) β mN,F := inf (Z G m ( N ) F ( x, P, H N ) dV : V ∈ HV m ( N ) , V = 0 with weak mean curvature H N relative to ¯ N ) and(3) α mN,F := inf (Z G m ( N ) F ( x, P, A ) dV : V ∈ CV m ( N ) , V = 0 with generalized second fundamental form A ) where F is as in Definition 2.2 and satisfies (44) ( respectively (38)). As the reader may see, theexpressions R G m ( N ) F ( x, P, H N ) dV (respectively R G m ( N ) F ( x, P, A ) dV ) are the natural generalizations ofthe functionals E pH,m (respectively E pA,m ) in (1) with p > m in the context of varifolds.Before stating the two main theorems, let us recall that an integral rectifiable m -varifold V on N isassociated with a “generalized m -dimensional subset” spt µ V of N together with an integer valued densityfunction θ ( x ) ≥ Theorem 1.1.
Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of the n -dimensional Riemannian manifold ( ¯ N , g ) isometrically embedded in some R S (by Nash Embedding The-orem), fix m ≤ n − and consider a function F : G m ( N ) × R S → R + satisfying (2.2) and (44) , namely F ( x, P, H ) ≥ C | H | p for some C > and p > m . hen, at least one of the following two statement is true:a) the space ( N, g ) contains a non zero m -varifold with null weak mean curvature H N relative to ¯ N (in other words, N contains a stationary m -varifold; see Remark 2.14 for the details),b) the minimization problem (2) corresponding to F has a solution i.e. there exists a non null integral m -varifold V ∈ HV m ( N ) with weak mean curvature H N relative to ¯ N such that Z G m ( N ) F ( x, P, H N ) dV = β mN,F = inf (Z G m ( N ) F ( x, P, ˜ H N ) d ˜ V : ˜ V ∈ HV m ( N ) , ˜ V = 0 ) . Moreover, in case b ) is true, we have β mN,F > and the minimizer V has the following properties:b1) the varifold V is indecomposable in HV m ( N ) b2) the diameter of spt µ V as a subset of the Riemannian manifold ( ¯ N , g ) is strictly positive diam ¯ N (spt µ V ) > . For the precise notion of indecomposability see Definition 2.15, intuitively (for more details see Remark2.16) the statement b
1) is telling that the support spt µ V of the spatial measure µ V associated to V isconnected. Remark 1.2.
It could be interesting to study the regularity of the minimizer V . Since H ∈ L p ( V ) , p > m given by (44) , we have the following structure result of Allard: fixed x ∈ spt µ V , under the hypothesis thatthe density in x satisfies θ ( x ) = 1 plus other technical assumptions (see Theorem 8.19 in [1]), spt µ V islocally a graph of a C , − mp function. Moreover, under similar assumptions, Duggan proved local W ,p regularity in [7]. In the multiple density case the regularity problem is more difficult. For instance, in[5], is given an example of a varifold ˜ V with bounded weak mean curvature whose spatial support containsa set C of strictly positive measure such that if x ∈ C then spt µ ˜ V does not correspond to the graph ofeven a multiple-valued function in any neighborhood of x . Nevertheless, recently, Menne proved that anintegral varifold with locally bounded first variation is C -rectifiable (see Theorem 1 in [17]). Now let us state the second main Theorem about the second fundamental form A . Theorem 1.3.
Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of the n -dimensional Riemannian manifold ( ¯ N , g ) isometrically embedded in some R S (by Nash Embedding The-orem), fix m ≤ n − and consider a function F : G m ( N ) × R S → R + satisfying (2.2) and (38) ,namely F ( x, P, A ) ≥ C | A | p for some C > and p > m .Then, at least one of the following two statements is true:a) the space ( N, g ) contains a non zero m -varifold with null generalized second fundamental form,b) the minimization problem (3) corresponding to F has a solution i.e. there exists a non null curvature m -varifold V ∈ CV m ( N ) with generalized second fundamental form A such that Z G m ( N ) F ( x, P, A ) dV = α mN,F = inf (Z G m ( N ) F ( x, P, ˜ A ) d ˜ V : ˜ V ∈ CV m ( N ) , ˜ V = 0 ) . Moreover, in case b ) is true, we have α mN,F > and the minimizer V has the following properties:b1) the varifold V is indecomposable in CV m ( N ) (see Definition 2.15),b2) the diameter of spt µ V as a subset of the Riemannian manifold ( ¯ N , g ) is strictly positive diam ¯ N (spt µ V ) > , b3) For every x ∈ spt µ V , V has a unique tangent cone at x and this tangent cone is a finite union of m -dimensional subspaces P i with integer multiplicities m i ; moreover, in some neighborhood of x we canexpress V has a finite union of graphs of C , − mp , m i -valued functions defined on the respective affinespaces x + P i ( p given in (38) ). emark 1.4. For the precise definitions and results concerning b , the interested reader can look atthe original paper [10] of Hutchinson. Notice that the boundary of N does not create problems since, byour definitions, the minimizer V is a fortiori an integral m -varifold with generalized second fundamentalform A ∈ L p ( V ) , p > m, in the n -dimensional Riemannian manifold ( ¯ N , g ) which has no boundary.Moreover, by Nash Embedding Theorem, we can assume ¯ N ⊂ R S ; therefore V can be seen as an integral m -varifold with generalized second fundamental form A ∈ L p ( V ) , p > m, in R S and the regularity theoremof Hutchinson can be applied.It could be interesting to prove higher regularity of the minimizer V . About this point, notice that itis not trivially true that V is locally a union of graphs of W ,p (Sobolev) functions. Indeed in [4] thereis an example of a curvature m -varifold ˜ V ∈ CV m ( R S ) , S ≥ , ≤ m ≤ S − , with second fundamentalform in L p , p > m , which is not a union of graphs of W ,p functions.In the spirit of proving higher regularity of the minimizer of such functionals we mention the preprintof Moser [24] where the author proves smoothness of the minimizers of R | A | in the particular case ofcodimension Lipschitz graphs in R S .Recall also the aforementioned result of Menne [17] giving that a curvature m -varifold is C -rectifiable. In both theorems, a delicate point is whether or not a ) is satisfied (fact which trivializes the result);we will study this problem in Section 7: we will recall two general classes of examples (given by Whitein [31]) of Riemannian manifolds with boundary where a ) is not satisfied in codimension 1, we will givetwo new examples for higher codimensions (namely Theorem 7.6 and Theorem 7.7) and we will proposea related open problem in Remark 7.9. Here, let us just remark that every compact subset N ⊂⊂ R S for s > a ) (see Theorem 7.7).The idea for proving the results is to consider a minimizing sequence { V k } k ∈ N of varifolds, show thatit is compact (i.e. there exists a varifold V and a subsequence { V k ′ } converging to V in an appropriatesense) and it is non degenerating: if the masses decrease to 0 the limit would be the null varifold sonot a minimizer, and if the diameters decrease to 0 the limit would be a point which has no geometricrelevance.In order to perform the analysis of the minimizing sequences, in Section 3 we prove monotonicityformulas for integral rectifiable m -varifolds in R S with weak mean curvature in L p , p > m . Theseformulas are similar in spirit to the ones obtained by Simon in [27] for smooth surfaces in R S involvingthe Willmore functional. These estimates are a fundamental tool for proving the non degeneracy of theminimizing sequences and we think they might have other applications.To show the compactness of the minimizing sequences it is crucial to have a uniform upper boundon the masses (for the non expert reader: on the volumes of the generalized submanifolds). Inspiredby the paper of White [31], in Section 4 we prove some isoperimetric inequalities involving our integralfunctionals which give the mass bound on the minimizing sequences in case a ) in the main theorems isnot satisfied. The compactness follows and is proved in the same Section. Also in this case, we thinkthat the results may have other interesting applications.The proofs of the two main theorems is contained in Section 5 and 6. Finally, as written above, Section7 is devoted to examples and remarks: we will notice that a large class of manifold with boundary canbe seen as compact subset of manifold without boundary, we will give examples where the assumptionfor the isoperimetric inequalities are satisfied and we will end with a related open problem.The new features of the present paper relies, besides the main theorems, in the new tools introducedin Section 3 and Section 4, and in the new examples presented in Section 7. Acknowledgments
This work has been supported by M.U.R.S.T under the Project FIRB-IDEAS “Analysis and Beyond”.The author would like to thank G. Bellettini, E. Kuwert, A. Malchiodi, C. Mantegazza, U. Menne andB. White for stimulating and fundamental discussions about the topics of this paper.4
Notations, conventions and basic concepts on varifolds
First of all, large positive constants are always denoted by C , and the value of C is allowed to varyfrom formula to formula and also within the same line. When we want to stress the dependence of theconstants on some parameter (or parameters), we add subscripts to C , as C N , etc.. Also constants withsubscripts are allowed to vary.In this Section we review the concept of curvature varifold introduced by Hutchinson in [9] giving aslightly more general definition; namely Hutchinson defines the curvature varifolds as “special” integralvarifolds in a Riemannian manifold but, as a matter of facts, the same definition makes sense for an evennon rectifiable varifold in a subset of a Riemannian manifold. So we will define (a priori non rectifiable)varifolds with curvature, which are endowed with a generalized second fundamental form.We start by recalling some basic facts about varifolds. For more details, the interested reader maylook at the standard references [8], [23], [28] or, for faster introductions, at [15] or the appendix of [31].Consider a (maybe non compact) n -dimensional Riemannian manifold ( ¯ N , g ). Without loss of gene-rality, by the Nash Theorem, we can assume that( ¯
N , g ) ֒ → R S isometrically embedded for some S > . We will be concerned with a subset N ⊂ ¯ N which, a fortiori, is also embedded in R S : N ֒ → R S . Sincethroughout the paper N ⊂⊂ ¯ N is a compact subset (in the end of the article we will also assume that ithas non empty interior int ( N ) = ∅ ) also in this Section is assumed to be so, even if most of the followingdefinitions and properties are valid for more general subsets.Let us denote with G ( S, m ) the Grassmaniann of unoriented m -dimensional linear subspaces of R S ,with G m ( ¯ N ) := ( R S × G ( S, m )) ∩ { ( x, P ) : x ∈ ¯ N , P ⊂ T x ¯ N m -dimensional linear subspace } and with G m ( N ) := G m ( ¯ N ) ∩ { ( x, P ) : x ∈ N } . We recall that an m - varifold V on N is a Radon measure on G m ( N ) and that the sequence of varifolds { V k } k ∈ N converges to the varifold V in varifold sense if V k → V weak as Radon measures on G m ( N ); i.e. Z G m ( N ) φ dV k → Z G m ( N ) φ dV as k → ∞ , for all φ ∈ C c ( G m ( N )). A special class of varifolds are the rectifiable varifolds : givena countably m -rectifiable, H m - measurable subset M of N ⊂ R S and θ a non negative locally H m integrable function on M , the rectifiable varifold V associated to M and θ is defined as V ( φ ) := Z M θ ( x ) φ ( x, T x M ) d H m ∀ φ ∈ C c ( G m ( N ))and sometimes is denoted with V ( M, θ ). Recall that if
M, θ are as above then the approximate tangentspace T x M exists for H m -almost every x ∈ M (Theorem 11.6 in [28], for the definitions see 11.4 of thesame book). If moreover θ is integer valued, then we say that V is an integral varifold ; the set of theintegral m -varifolds in N is denoted by IV m ( N ).If V is a k -varifold, let | V | denote its mass: | V | := V ( G m ( N )) . Observe that we have a natural projection(4) π : G m ( N ) → N ( x, P ) x, V via the projection π , we have a positive Radon measure µ V on Nµ V ( B ) := V ( π − ( B )) = V ( G m ( B )) ∀ B ⊂ N Borel set . Since V is a measure on G m ( N ), its support is a closed subset of G m ( N ). If we project that closed seton N by the projection π then we get the spatial support of V , which coincides with spt µ V .Now let us define the notion of measure-function pair. Definition 2.1.
Let V be a Radon measure on G m ( N ) (i.e. a varifold) and f : G m ( N ) → R α be a welldefined V almost everywhere L loc ( V ) function. Then we say that ( V, f ) is a measure-function pair over G m ( N ) with values in R α .Given { ( V k , f k ) } k ∈ N and ( V, f ) measure-function pairs over G m ( N ) with values in R α , suppose V k → V weak as Radon measures in G m ( N ) (or equivalently as varifolds in N ). Then we say ( V k , f k ) convergesto ( V, f ) in the weak sense and write ( V k , f k ) ⇀ ( V, f ) if V k ⌊ f k → V ⌊ f weak convergence of Radon vector valued measures. In other words, if Z G m ( N ) h f k , φ i dV k → Z G m ( N ) h f, φ i dV as k → ∞ , for all φ ∈ C c ( G m ( N ) , R α ) , where h ., . i is the scalar product in R α . Definition 2.2.
Suppose F : G m ( N ) × R α → R . We will denote the variables in G m ( N ) × R α by (x,P,q).We say that F satisfies the condition (2.2) if the following statements are verified:i) F is continuous,ii) F is non negative ( i.e. F ( x, P, q ) ≥ for all ( x, P, q ) ∈ G m ( N ) × R α ) and F ( x, P, q ) = 0 if and onlyif q = 0 ,iii) F is convex in the q variables, i.e. F ( x, P, λq + (1 − λ ) q ) ≤ λF ( x, P, q ) + (1 − λ ) F ( x, P, q ) for all λ ∈ (0 , , ( x, P ) ∈ G m ( N ) , q ∈ R α , q ∈ R α ,iv) F has superlinear growth in the q variables, i.e. there exists a continuous function φ , where φ : G m ( N ) × [0 , ∞ ) → [0 , ∞ ) , ≤ φ ( x, P, s ) ≤ φ ( x, P, t ) for ≤ s ≤ t and ( x, P ) ∈ G m ( N ) , φ ( x, P, t ) → ∞ locally uniformly in ( x, P ) as t → ∞ , such that φ ( x, P, | q | ) | q | ≤ F ( x, P, q ) for all ( x, P, q ) ∈ G m ( N ) × R α . An example (trivial but fundamental for this paper) of such an F is F ( x, P, q ) := | q | p for any p > Remark 2.3.
For simplicity, in Definition 2.2, we assumed the same conditions of Hutchinson ([9]Definition 4.1.2) on F but some hypotheses can be relaxed. For example, about the results in this paper,if F = F ( q ) depends only on the q variables it is enough to assume (in place of i)) that F is lowersemicontinuous (see Theorem 6.1 in [16]). In the aforementioned paper, Hutchinson proves the following useful compactness and lower semicon-tinuity Theorem (see Theorem 4.4.2 in [9]):
Theorem 2.4.
Suppose { ( V k , f k ) } k ∈ N are measure-function pairs over G m ( N ) with values in R α . Sup-pose V is a Radon measure on G m ( N ) (i.e a varifold in N ) and V k → V weak converges as Radonmeasures (equivalently varifold converges in N ). Suppose F : G m ( N ) × R α → R satisfies the condition (2.2) . Then the following are true:i) If there exists C > such that (5) Z G m ( N ) F ( x, P, f k ( x, P )) dV k ≤ C ∀ k ∈ N hen there exists a function f ∈ L loc ( V ) such that, up to subsequences, ( V k , f k ) ⇀ ( V, f ) .ii) if there exists C > such that (5) is satisfied and ( V k , f k ) ⇀ ( V, f ) then Z G m ( N ) F ( x, P, f ( x, P )) dV ≤ lim inf k Z G m ( N ) F ( x, P, f k ( x, P )) dV k . Now we want to define the varifolds of N with curvature. Observe that given ( x, P ) ∈ G m ( N ), the m -dimensional linear subspace P ⊂ T x ¯ N ⊂ R S can be identified with the orthogonal projection matrixon Hom ( R S , R S ) ∼ = R S P ≡ [ P ij ] ∈ R S . Similarly, the tangent space of ¯ N at x can be identified with its orthogonal projection matrix T x ¯ N ≡ Q ( x ) := [ Q ij ( x )] ∈ R S . Before defining the varifolds with curvature let us introduce a bit of notation: given φ = φ ( x, P ) ∈ C ( R S × R S ) we call the partial derivatives of φ with respect to the variables x i and P jk (freezing allother variables) by D i φ and D ∗ jk φ for i, j, k = 1 , . . . , S respectively. In the following definition we will consider the quantity P ij ∂ψ∂x j ( x ) for ψ ∈ C ( ¯ N );we mean that ψ is extended to a C function to some neighborhood of x ∈ R S and, since P is theprojection on a m -subspace of T x ¯ N , the definition does not depend on the extension. Observe moreoverthat the quantity depends on ( x, P ) so it is a function on G m ( ¯ N ). Definition 2.5.
Let V be an m -varifold on N ⊂ ¯ N ֒ → R S , m ≤ n − . We say that V is a varifold with(generalized) curvature or with (generalized) second fundamental form if there exist real-valued functions B ijk , for ≤ i, j, k ≤ S , defined V almost everywhere in G m ( N ) such that on setting B = [ B ijk ] thefollowing are true:i) ( V, B ) is a measure-function pair over G m ( N ) with values in R S .ii) For all functions φ = φ ( x, P ) ∈ C c ( R S × R S ) one has (6) 0 = Z G m ( N ) [ P ij D j φ + B ijk D ∗ jk φ + B jij φ ] dV for i = 1 , . . . , S. In this case B is called (generalized) curvature and we can also define the (generalized) second funda-mental form of V (with respect to ¯ N ) as the L loc ( V ) function with values in R S A : G m ( N ) → R S ,A kij ( x, P ) := P lj B ikl ( x, P ) − P lj P iq ∂Q kl ∂x q ( x ) . (7) We will denote the set of integral m -varifolds of N with generalized curvature as CV m ( N ) and we willcall them curvature m -varifolds . A varifold V is said to have null generalized second fundamental form if there exists a generalized curvature B whose associated generalized second fundamental form A is null. Observe that we use different notation of [9]: we call B what Hutchinson calls A and vice versa; this isbecause we want to denote with A the second fundamental form with respect to ¯ N . Moreover, as shownin Section 5 of [9], if V is the integral varifold associated to a smooth immersed m -submanifold of N then A coincides with the classical second fundamental form with respect to N .7 emark 2.6. By definition, the generalized second fundamental form A is expressed in terms of B but,as Hutchinson proved in [9] Propositions 5.2.4 and 5.2.6, it is possible to express B in terms of A . Indeed,choosing appropriate test functions, with some easy computations one can prove that (8) B ijk = A kij + A jik + P jl P iq ∂Q lk ∂x q ( x ) + P kl P iq ∂Q lj ∂x q ( x ) . Remark 2.7.
As U. Menne pointed out to the author in a personal communication, the proof of Hutchin-son of the uniqueness of the generalized curvature B (see [9], Proposition 5.2.2) uses in a crucial waythat, in his case, the varifold is rectifiable.At the level of generality of Definition 2.5, i.e. V is an a priori non rectifiable varifold, the argumentsof Hutchinson does not ensure the uniqueness of the generalized second fundamental form, and to ourknowledge the uniquness is open in the general case.However throughout the paper we mostly work with integral varifolds, for which the uniqueness of thesecond fundamental form is granted by [9]; the only place where we need a priori non rectifiable varifoldswith generalized second fundamental form is in Theorem 4.1 and its applications, where we use the conceptof maybe non-rectifiable varifold with null generalized second fundamental form. We will mean a varifold V such that there exists a generalized curvature B whose associated generalized second fundamental form A , as in (7) , is null. Now let us recall the fundamental compactness and lower semi continuity Theorem of Hutchinson(Theorem 5.3.2 in [9])
Theorem 2.8.
Consider { V k } k ∈ N ⊂ CV m ( N ) with generalized second fundamental forms { A k } k ∈ N , V an integral m -varifold of N and suppose V k → V in varifold sense. Let F : G m ( N ) × R S → R be afunction satisfying the condition (2.2) and assume that Z G m ( N ) F ( x, P, A k ) dV k ≤ C for some C > independent on k . Theni) V ∈ CV m ( N ) with generalized second fundamental form A ,ii) ( V k , A k ) ⇀ ( V, A ) in the weak sense of measure-function pairs,iii) R G m ( N ) F ( x, P, A ) dV ≤ lim inf k R G m ( N ) F ( x, P, A k ) dV k . Now we briefly recall the definition of first variation of an m -varifold V in R S ; the original definitionsare much more general, here we recall only the facts we need for this paper. Definition 2.9.
Let V be an m -varifold in R S and let X be a C c ( R S ) vector field. We define firstvariation δV the linear functional on C c ( R S ) vector fields δV ( X ) := Z G m ( R S ) div P X ( x ) dV ( x, P ); where for every P ∈ G ( S, m ) , div P X := S X i =1 ∇ Pi X i = S X i,j =1 P ij D j X i , where ∇ P f = P ( ∇ f ) is the projection on P of the gradient in R S of f and ∇ Pi := e i · ∇ P (where { e i } i =1 ,...,S is an orthonormal basis of R S ). V is said to be of locally bounded first variation in R S if for every relatively compact open W ⊂⊂ R S there exists a constant C W < ∞ such that | δV ( X ) | ≤ C W sup W | X | for all X ∈ C c ( R S ) vector fields with support in W .
8n interesting subclass of varifolds with locally bounded first variation are the varifolds with weakmean curvature.
Definition 2.10.
Let V be an m -varifold in R S and H : G m ( R S ) → R S an L loc ( V ) function (in theprevious notation we would say that ( V, H ) is a measure-function pair on G m ( R S ) with values in R S );then we say that V has weak mean curvature H if for any vector field X ∈ C c ( R S ) one has (9) δV ( X ) := Z G m ( R S ) div P X ( x ) dV ( x, P ) = − Z G m ( R S ) H · XdV ( x, P ) . Observe that if V = V ( M, θ ) is a rectifiable varifold with weak mean curvature H then with abuse ofnotation we can write H ( x ) = H ( x, T x M ) and we get the following identities:(10) Z M div M Xdµ V = Z G m ( R S ) div T x M X ( x ) dV = − Z G m ( R S ) H ( x, T x M ) · XdV = − Z M H ( x ) · Xdµ V , where div M X is the tangential divergence of the vector field X and is defined to be div M X ( x ) := div T x M X ( x ) where T x M is the approximate tangent space to M at x (which exists for µ V -a.e. x ). Remark 2.11.
As Hutchinson observed in [9], if V is an m -varifold on N ֒ → R S with generalizedcurvature B = [ B ijk ] i,j,k =1 ,...,S then, as a varifold in R S , V has weak mean curvature H i = P Sj =1 B jij for i = 1 , . . . , S . Indeed, for any relatively compact open subset W ⊂⊂ R S and any vector field X ∈ C c ( R S ) with compact support in W , taking φ = X i , i = 1 , . . . , S in equation (6) and summing over i we get Z G m ( R S ) [ P ij D j X i ( x ) + B jij ( x, P ) X i ( x )] dV ( x, P ) which implies δV ( X ) := Z G m ( R S ) div P X ( x ) dV ( x, P ) = − Z G m ( R S ) B jij ( x, P ) X i ( x ) dV ( x, P ); the conclusion follows from the fact that B ∈ L loc ( V ) . Now let us define the varifolds with weak mean curvature in a compact subset N ⊂⊂ ¯ N of a Rieman-nian manifold ( N, g ) isometrically embedded in R S . Definition 2.12.
Let V be an m -varifold on N ⊂ ¯ N ֒ → R S , m ≤ n − . We say that V is a varifoldwith weak mean curvature H N relative to ¯ N if it has weak mean curvature H R S as varifold in R S . Inthis case the value of ( H N ) i , i = 1 , . . . , S is given by (11) ( H N ) i = ( H R S ) i − P jk ∂Q ij ∂x k . Consistently with the notation introduced for the curvature varifolds, we denote with HV m ( N ) the set of integral m -varifolds on N with weak mean curvature relative to ¯ N ; the elements of HV m ( N ) are called mean curvature varifolds . Observe that in case V is the varifold associated to a smooth submanifold of ¯ N then H N coincideswith the classical mean curvature relative to ¯ N (it is enough to trace the identity (i) of Proposition 5.1.1in [9] recalling that we denote with A , Q what Hutchinson calls B , S ). Moreover, as an exercise, the readermay check that also in the general case the vector (cid:16) P jk ∂Q ij ∂x k (cid:17) i =1 ,...,S of R S is orthogonal to ¯ N (fix a point x of ¯ N and choose a base of T x ¯ N in which the Christoffel symbols of ¯ N vanish at x ; write down theorthogonal projection matrix Q with respect to this base and check the orthogonality condition).9 emark 2.13. If V is an m -varifold on N ⊂ ¯ N ֒ → R S , m ≤ n − with weak mean curvature H N relative to ¯ N then, for each compactly supported vector field X ∈ C c ( ¯ N ) tangent to ¯ N , δV ( X ) = Z G m ( N ) div P X ( x ) dV ( x, P ) = − Z G m ( N ) H N · XdV ( x, P ) . This fact gives consistency to Definition 2.12 and follows from Definition 2.12, from formula (9) andthe orthogonality of (cid:16) P jk ∂Q ij ∂x k (cid:17) i =1 ,...,S to ¯ N . Remark 2.14. If V is an m -varifold on N ⊂ ¯ N ֒ → R S , m ≤ n − with null weak mean curvature H N = 0 relative to ¯ N then, for each compactly supported vector field X ∈ C c ( ¯ N ) tangent to ¯ N , δV ( X ) = Z G m ( N ) div P X ( x ) dV ( x, P ) = 0 . In this case we say that V is an m -varifold in N with null weak mean curvature relative to ¯ N or, usingmore classical language, that V is a stationary m -varifold in N (where stationary as to be intended in ¯ N ). Definition 2.15.
A curvature m -varifold V ∈ CV m ( N ) (resp. V ∈ HV m ( N ) ) is said to be decomposablein CV m ( N ) (resp. in HV m ( N ) ) if there exist two non null curvature varifolds = V , V ∈ CV m ( N ) (resp. = V , V ∈ HV m ( N ) ) such that V = V + V . If V is not decomposable in CV m ( N ) (resp. HV m ( N ) ) itis said indecomposable in CV m ( N ) (resp. in HV m ( N ) ). Remark 2.16.
Notice that, given V ∈ CV m ( N ) (resp. V ∈ HV m ( N ) ), if the support of the spatial mea-sure spt µ V has two connected components at positive distance then V is decomposable; indeed, localizingDefinition 2.5 using cutoff functions, it is clear that the two connected components detect two non nullelements of CV m ( N ) whose sum is the original varifold V .In case instead spt µ V has countably many connected components which are accumulating, it is not soclear if each connected component detect an element of CV m ( N ) (resp. HV m ( N ) ) as it is not possible toisolate each component by using cutoff functions.On the other hand even if spt µ V is connected the varifold V may be decomposable, as the example of twosmooth embedded compact submanifolds with non empty intersection shows. m -varifolds with weak meancurvature in L p , p > m Let V = V ( M, θ ) be an integral varifold of R S (associated to the rectifiable set M ⊂ R S and with integermultiplicity function θ ) with weak mean curvature H (since throughout this section we consider onlyvarifolds in R S and there is no ambiguity, we adopt the easier notation H for H R S ). Let us write µ for µ V := π ♯ ( V ) the push forward of the varifold measure V on G m ( N ) to N via the standard projection π : G m ( N ) → N, π ( x, P ) = x (see Section 2 for more details); of course µ V can also be seen as µ V = H m ⌊ θ ,the restriction of the m -dimensional Hausdorff measure to the multiplicity function θ .The first Lemma is a known fact (see for example the book of Leon Simon [28] at page 82) of whichwe report also the proof for completeness. Lemma 3.1.
Let V = V ( M, θ ) ∈ IV m ( R S ) be with weak mean curvature H as above and fix a point x ∈ M . For µ -a.e. x ∈ M call r ( x ) := | x − x | and D ⊥ r the orthogonal projection of the gradient vector Dr onto ( T x M ) ⊥ . Consider a nonnegative function φ ∈ C ( R ) such that φ ′ ( t ) ≤ ∀ t ∈ R , φ ( t ) = 1 for t ≤ , φ ( t ) = 0 for t ≥ . or all ρ > let us denote I ( ρ ) := Z M φ ( r/ρ ) dµ,L ( ρ ) := Z M φ ( r/ρ )( x − x ) · Hdµ,J ( ρ ) := Z M φ ( r/ρ ) | D ⊥ r | dµ ; then (12) ddρ [ ρ − m I ( ρ )] = ρ − m J ′ ( ρ ) + ρ − m − L ( ρ ) . Proof . The idea is to use formula (10) and choose the vector field X in an appropriate way in orderto get informations about V . First of all let us recall that for any function f ∈ C ( R S ) and any x ∈ M where the approximate tangent space T x M exists (it exists for µ -a.e. x ∈ M see [28] 11.4-11.6 ) one candefine the tangential gradient as the projection of the gradient in R S onto T x M : ∇ M f := S X j,l =1 P jl D l f ( x ) e j where D l f denotes the partial derivative ∂f∂x l of f , P jl is the matrix of the orthogonal projection of R S onto T x M and { e j } j =1 ,...,S is an orthonormal basis of R S . Denoted ∇ Mj := e j · ∇ M , recall that thetangential divergence is defined as div M X := S X j =1 ∇ Mj X j ;moreover it is easy to check the Leibniz formula div M f X := ∇ M f · X + f div M X ∀ f ∈ C ( R S ) and ∀ X ∈ C ( R S ) vector field.Now let us choose the vector field. Fix ρ > γ ∈ C ( R ) defined as γ ( t ) := φ ( t/ρ );then of course we have the following properties: γ ′ ( t ) ≤ ∀ t ∈ R , γ ( t ) = 1 for t ≤ ρ , γ ( t ) = 0 for t ≥ ρ. Call r ( x ) := | x − x | and choose the vector field X ( x ) := γ ( r ( x ))( x − x ) . Using the Leibniz formula we get div M X = ∇ M γ ( r ) · ( x − x ) + γ ( r ) div M ( x − x )= rγ ′ ( r ) ( x − x ) T | x − x | ( x − x ) T | x − x | + mγ ( r )= rγ ′ ( r )(1 − | D ⊥ r | ) + mγ ( r ) , (13)where u T is the projection of the vector u ∈ R S onto T p M and D ⊥ r = ( x − x ) ⊥ | x − x | is the orthogonal projectionof the gradient vector Dr onto ( T x M ) ⊥ . The equation (10) of the weak mean curvature thus yields(14) m Z M γ ( r ) dµ + Z M rγ ′ ( r ) dµ = Z M rγ ′ ( r ) | D ⊥ r | dµ − Z M H · ( x − x ) γ ( r ) dµ. γ ( r ) = φ ( r/ρ ), so rγ ′ ( r ) = rρ φ ′ ( r/ρ ) = − ρ ∂∂ρ [ φ ( r/ρ )] . Thus, combining (14) and thedefinitions of I ( ρ ) , J ( ρ ) and L ( ρ ) one gets mI ( ρ ) − ρI ′ ( ρ ) = − ρJ ′ ( ρ ) − L ( ρ ) . Thus, multiplying both sides by ρ − m − and rearranging we obtain ddρ [ ρ − m I ( ρ )] = ρ − m J ′ ( ρ ) + ρ − m − L ( ρ ) . This concludes the proofEstimating from below the right hand side of (12) and integrating, we get the following useful ine-qualities.
Proposition 3.2.
Let V = V ( M, θ ) ∈ IV m ( R S ) be with weak mean curvature H ∈ L p ( V ) , p > m (wemean that R G m ( R S ) | H | p dV < ∞ or equivalently, denoted with an abuse of notation H ( x ) = H ( x, T x M ) , R M | H | p dµ < ∞ ). Fixed a point x ∈ M and < σ < ρ < ∞ , then (15)[ σ − m µ ( B σ ( x ))] p ≤ [ ρ − m µ ( B ρ ( x ))] p + p p − m ρ − mp (cid:16) Z B ρ ( x ) | H | p dµ (cid:17) p − p p − m σ − mp (cid:16) Z B σ ( x ) | H | p dµ (cid:17) p . Proof . Let us estimate from below the right hand side of equation (12). Observe that J ′ ( ρ ) = ddρ Z M φ ( r/ρ ) | D ⊥ r | dµ = − ρ − Z M rφ ′ ( r/ρ ) | D ⊥ r | dµ ≥ φ ′ ( t ) ≤ t ∈ R . Thus we can say that(16) ddρ [ ρ − m I ( ρ )] ≥ ρ − m − L ( ρ ) . Let us estimate from below the right hand side by Schwartz inequality: ρ − m − L ( ρ ) = ρ − m − Z M φ ( r/ρ )( x − x ) · Hdµ ≥ − ρ − m − Z M (cid:0) φ ( r/ρ ) p | H | (cid:1) | x − x | φ ( r/ρ ) p − p dµ. Now recalling that φ ( t ) = 0 for t ≥ φ ( r/ρ ) = 0 for r ≥ ρ so | x − x | in the integral can beestimated from above by ρ and we can say that ρ − m − L ( ρ ) ≥ − ρ − m Z M (cid:0) φ ( r/ρ ) p | H | (cid:1) φ ( r/ρ ) p − p dµ ;thus, by Holder inequality, for all p > ρ − m − L ( ρ ) ≥ − ρ − m (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p (cid:16) Z M φ ( r/ρ ) dµ (cid:17) p − p = − ρ − m (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p I ( ρ ) p − p . (17)Putting together inequalities (16) and (17) we get ddρ [ ρ − m I ( ρ )] ≥ − ρ − m (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p I ( ρ ) p − p ;12ultiplying both sides by ρ m − mp I ( ρ ) p − and rearranging we get ddρ [ ρ − m I ( ρ )] p ≥ − p ρ − mp (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p . Now, after choosing p > m , integrate the last inequality from σ to ρ (the same ρ chosen in the statementof the Proposition) and get with an integration by parts of the right hand side ρ − mp I ( ρ ) p − σ − mp I ( σ ) p ≥ − p Z ρσ h (cid:16) t − mp (cid:17) (cid:16) Z M φ ( r/t ) | H | p dµ (cid:17) p i dt = − p h(cid:16) − mp (cid:17) − (cid:16) ρ − mp (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p − σ − mp (cid:16) Z M φ ( r/σ ) | H | p dµ (cid:17) p (cid:17)i + p Z ρσ h(cid:16) − mp (cid:17) − t − mp (cid:16) ddt Z M φ ( r/t ) | H | p dµ (cid:17)i dt (18)Observe that, as before for J ′ ( ρ ), since φ ′ ( t ) ≤ t it follows ddt Z M φ ( r/t ) | H | p dµ = − t − Z M rφ ′ ( r/t ) | H | p dµ ≥ I , we can write (cid:16) ρ − m Z M φ ( r/ρ ) dµ (cid:17) p − (cid:16) σ − m Z M φ ( r/σ ) dµ (cid:17) p ≥ p p − m h − ρ − mp (cid:16) Z M φ ( r/ρ ) | H | p dµ (cid:17) p + σ − mp (cid:16) Z M φ ( r/σ ) | H | p dµ (cid:17) p i . (19)Now observe that during all this proof and during all the proof of Lemma 3.1 the only used properties of φ have been φ ∈ C ( R ) , φ ′ ( t ) ≤ ∀ t ∈ R , φ ( t ) ≤ ∀ t ∈ R , φ ( t ) = 0 ∀ t ≥ φ , the inequality (19) holds. Now taking a sequence φ k of such functions pointwiseconverging to the characteristic function of ] − ∞ ,
1] and, using the Dominated Convergence Theorem,passing to the limit on k in (19) we get (cid:2) ρ − m µ ( B ρ ( x )) (cid:3) p − (cid:2) σ − m µ ( B σ ( x )) (cid:3) p ≥ p p − m h − ρ − mp (cid:16) Z B ρ ( x ) | H | p dµ (cid:17) p + σ − mp (cid:16) Z B σ ( x ) | H | p dµ (cid:17) p i . Rearranging we can conclude that (cid:2) σ − m µ ( B σ ( x )) (cid:3) p ≤ (cid:2) ρ − m µ ( B ρ ( x )) (cid:3) p + p p − m ρ − mp (cid:16) Z B ρ ( x ) | H | p dµ (cid:17) p − p p − m σ − mp (cid:16) Z B σ ( x ) | H | p dµ (cid:17) p From Corollary 17.8 page 86 of [28], if H ∈ L p ( V ) for some p > m , then the density θ ( x ) =lim ρ ↓ µ ( ¯ B ρ ( x )) w m ρ m exists at every point x ∈ R S and θ is an upper semicontinuous function. Hence, let-ting σ →
0, one has [ ω m θ ( x )] p ≤ h µ ( B ρ ( x )) ρ m i p + p p − m h ρ p − m Z B ρ ( x ) | H | p dµ i p . Using the inequality a p + b p ≤ p − p ( a + b ) p given by the concavity of the function t t p with p > t >
0, we get ω m θ ( x ) ≤ p − h µ ( B ρ ( x )) ρ m + (cid:16) p p − m (cid:17) p ρ p − m Z B ρ ( x ) | H | p dµ i . V ∈ IV m ( R S ), then θ is integer valued and by definition θ ≥ µ -a.e. From the upper semicontinuityof θ it follows that θ ( x ) ≥ x ∈ spt µ (where, as before, µ is the spatial measure associated to V ).Then the last formula can be written more simply getting the fundamental inequality (20) 1 ≤ C p,m h µ ( B ρ ( x )) ρ m + ρ p − m Z B ρ ( x ) | H | p dµ i ∀ x ∈ spt µ, where C p,m > p, m and such that C p,m → ∞ if p ↓ m . Remark 3.3.
The fundamental inequality can be extended to the case p = m by using the isoperimetricinequality, see 8.3 in [1]; for discussions on related results see Proposition 3.1 in [14] and 2.5 in [18]. Using the fundamental inequality now we can link through inequalities the mass of V , the diameterof M and the L p norm of the weak mean curvature H . Lemma 3.4.
Let V = V ( M, θ ) ∈ IV m ( R S ) be a non null integral m -varifold with compact spatial support spt µ ⊂ R S and weak mean curvature H ∈ L p ( V ) for some p > m . Then, called d = diam R S (spt µ ) thediameter of spt µ as a subset of R S , (21) | V | ≤ (cid:18) dm (cid:19) p Z M | H | p dµ. Proof . In the same spirit of the proof of Lemma 3.1 we choose a suitable vector field X to plug in themean curvature equation (10) Z M div M Xdµ = − Z M X · Hdµ in order to get informations about the varifold V = V ( M, θ ). Now fix a point x ∈ spt µ and simply let X ( x ) = x − x . Since div M X = m µ -a.e. (for more details see the proof of Lemma 3.1), observing that | X | ≤ d µ -a.e. and estimating the right hand side by Holder inequality we get m | V | ≤ d (cid:16) Z M | H | p dµ (cid:17) p | V | p − p . Now multiply both sides by | V | p − and raise to the power p in order to get the thesis. Lemma 3.5.
Let V = V ( M, θ ) ∈ IV m ( R S ) be a non null integral indecomposable (in HV m ( R S ) ) m -varifold with compact spatial support spt µ ⊂ R S and weak mean curvature H ∈ L p ( V ) for some p > m .Then, called d = diam R S (spt µ ) , (22) d ≤ C p,m (cid:16) Z M | H | p dµ (cid:17) m − p | V | − m − p where C p,m > is a positive constant depending on p, m and such that C p,m → ∞ if p ↓ m . Proof . Since spt µ ⊂ R S is compact, then there exist x , y ∈ spt µ such that d = | x − y | . Let ρ ∈ ]0 , d/
2] and call N := ⌊ d/ρ ⌋ the integer part of d/ρ . For every j = 1 , . . . , N − V implies the existence of r j ∈ (cid:0) j + , j + (cid:1) such that B r j ρ ( y ) ∩ spt µ = ∅ . y j ∈ ∂B r j ρ ( y ) ∩ spt µ and observe that for each ball B ρ/ ( y j ) , j = 0 , . . . , N − B ρ/ ( y j ) , j = 0 , . . . , N − j we get N ≤ C p,m (cid:18) | V | ρ m + ρ p − m Z M | H | p dµ (cid:19) . Moreover, since N = ⌊ d/ρ ⌋ ≥ d ρ , we have(23) d ≤ ρN ≤ C p,m (cid:18) | V | ρ m − + ρ p − m +1 Z M | H | p dµ (cid:19) . Now let us choose ρ in an appropriate way; observe that taken ρ = m (cid:18) | V | R M | H | p dµ (cid:19) p , in force of the estimate (21), the condition ρ ≤ d/ ρ intoequation (23), after some trivial computation we conclude that d ≤ C p,m | V | p − m +1 p (cid:18)Z M | H | p dµ (cid:19) m − p . Remark 3.6.
Notice that, in Lemma 3.5, the assumption that V is indecomposable in HV m ( R S ) can bereplaced by asking that the support of the spatial measure spt µ is connected. Combining the Fundamental Inequality with the previous lemmas we are in position to prove a lowerdiameter and mass bound.
Lemma 3.7.
Let V = V ( M, θ ) ∈ IV m ( R S ) be a non null integral m -varifold with spatial support spt µ ⊂ R S and weak mean curvature H ∈ L p ( V ) for some p > m . Then, called d := diam R S (spt µ )(24) d ≥ C p,m (cid:16) R M | H | p dµ (cid:17) p − m where C p,m > is a positive constant depending on p, m and such that C p,m → ∞ if p ↓ m . Proof . If d = ∞ , the inequality (24) is trivially satisfied; hence we can assume that spt µ ⊂ R S iscompact. It follows that there exist x , y ∈ spt µ such that d = | x − y | . Recall the Fundamental Inequality (20) and choose ρ = d obtaining(25) 1 ≤ C p,m (cid:16) | V | d m + d p − m Z M | H | p dµ (cid:17) . From Lemma 3.4, | V | ≤ m p d p Z M | H p | dµ, hence the inequality (25) becomes 1 ≤ C p,m d p − m Z M | H | p dµ and we can conclude. 15 emma 3.8. Let V = V ( M, θ ) ∈ IV m ( R S ) be a non null integral m -varifold with compact spatial support spt µ ⊂ R S and weak mean curvature H ∈ L p ( V ) for some p > m . Then (26) | V | ≥ C p,m (cid:16) R M | H | p dµ (cid:17) mp − m where C p,m > is a positive constant depending on p, m and such that C p,m → ∞ if p ↓ m . Proof . First of all if V is decomposable in HV m ( R S ) then, by definition, each component is an integralvarifold with weak mean curvature in L p . Hence we can assume that V is indecomposable in HV m ( R S ),otherwise just argue on a non null component and observe that the inequality (26) is well behaved.Call as before d := diam R S (spt µ ); from the inequality (22), | V | ≥ d pp − m +1 (cid:0)R M | H | p dµ (cid:1) m − p − m +1 . But from the last inequality (24), d pp − m +1 ≥ C p,m (cid:16) R M | H | p dµ (cid:17) p ( p − m )( p − m +1) . Combining the two estimates, with an easy computation we get the conclusion.
Proposition 3.9.
Let { V k = V k ( M k , θ k ) } k ∈ N ⊂ IV m ( R S ) be a sequence of integral varifolds with weakmean curvature H k ∈ L p ( V k ) for some p > m and associated spatial measures µ k . Assume a uniformbound on the L p norms of H k : ∃ C > ∀ k ∈ N Z M k | H k | p dµ k = Z G m ( R S ) | H k | p dV k ≤ C, and assume a uniform bound on the spatial supports spt µ k : ∃ R > µ k ⊂ B R S R where B R S R is the ball of radius R centered in the origin in R S .It follows that if there exists a Radon measure µ on R S such that µ k → µ weak as Radon measures,then spt µ k → spt µ in Hausdorff distance sense. Proof . First of all observe that the uniform bound on the spatial supports spt µ k implies that spt µ iscompact. Since spt µ is compact, recall that spt µ k → spt µ if and only if the set of the all possible limitpoints of all possible sequences { x k } k ∈ N with x k ∈ spt µ k coincides with spt µ . Let us prove it by doubleinclusion.i) since µ k → µ weak as Radon measures of course ∀ x ∈ spt µ there exists a sequence { x k } k ∈ N with x k ∈ spt µ k such that x k → x . Otherwise there would exist ǫ > k ′ B ǫ ( x ) ∩ spt µ k ′ = ∅ . This would imply that µ k ′ ( B ǫ ( x )) = 0, but x ∈ spt µ so we reach the contradiction0 < µ ( B ǫ ( x )) = lim k ′ µ k ′ B ǫ ( x ) = 0 . { x k } k ∈ N with x k ∈ spt µ k be such that x k → x . We have to show that x ∈ spt µ . Let us argueby contradiction:if x / ∈ spt µ then there exists ǫ > µ ( B ǫ ( x )) = lim k µ k ( B ǫ ( x )) . Since spt µ k ∋ x k → x , then for every ǫ ∈ (0 , ǫ /
2) there exists K ǫ > x k ∈ (spt µ k ∩ B ǫ ( x )) ∀ k > K ǫ . Now consider the balls B ǫ ( x k ) for k > K ǫ : by the triangle inequality B ǫ ( x k ) ⊂ B ǫ ( x ), moreover, sinceby construction x k ∈ spt µ k , we can apply the fundamental inequality (20) to each B ǫ ( x k ) and obtain1 ≤ C p,m h µ k ( B ǫ ( x k )) ǫ m + ǫ p − m Z B ǫ ( x k ) | H k | p dµ k i ≤ C p,m h µ k ( B ǫ ( x )) ǫ m + ǫ p − m Z M k | H k | p dµ k i ∀ k > K ǫ . (28)Keeping in mind (27), for every fixed ǫ ∈ (0 , ǫ /
2) we can pass to the limit on k in inequality (28) andget lim inf k Z M k | H k | p dµ k ≥ C p,m ǫ p − m . But ǫ > R M k | H k | p dµ k ≤ C of the assumptions. The following Isoperimetric Inequality involving the generalized second fundamental form is inspired bythe paper of White [31] and uses the concept of varifold with second fundamental form introduced byHutchinson [9]. Actually we need a slight generalization of the definition of curvature varifold given byHutchinson: in Definition 5.2.1 of [9], the author considers only integral varifolds but, as a matter offacts, a similar definition makes sense for a general varifold. In Section 2 we recalled the needed concepts.
Theorem 4.1.
Let N ⊂⊂ ¯ N be a compact subset of a (maybe non compact) n -dimensional Riemannianmanifold ( ¯ N , g ) (which, by Nash Embedding Theorem we can assume isometrically embedded in some R S )and let m ≤ n − . Then the following conditions are equivalent:i) N contains no nonzero m -varifold with null generalized second fundamental formii) There is an increasing function Φ : R + → R + with Φ(0) = 0 and a function F : G m ( N ) × R S → R + satisfying (2.2) (see Section 2) such that for every m -varifold V in N with generalized second fundamentalform A | V | ≤ Φ (cid:16) Z G m ( N ) F ( x, P, A ( x, P )) dV (cid:17) . iii) for every function F : G m ( N ) × R S → R + satisfying (2.2) (see Section 2) there exists a constant C F > such that for every m -varifold V in N with generalized second fundamental form A | V | ≤ C F Z G m ( N ) F ( x, P, A ( x, P )) dV. roof . Of course iii) ⇒ ii) ⇒ i). It remains to prove that i) ⇒ iii). Let us argue by contradiction:assume that iii) is not satisfied and prove that also i) cannot be satisfied.First fix the function F . If iii) is not satisfied then there exists a sequence { ( V k , A k ) } k ∈ N of m -varifoldsin N with generalized second fundamental form (see Definition 2.5) such that | V k | ≥ k Z G m ( N ) F ( x, P, A k ( x, P )) dV k . We can assume that | V k | = 1 otherwise replace V k with the normalized varifold ˜ V k := | V k | V k (ob-serve that the second fundamental form is invariant under this rescaling of the measure and that R G m ( N ) F ( x, P, A k ) dV k = | V k | R G m ( N ) F ( x, P, A k ) d ˜ V k ). Hence Z G m ( N ) F ( x, P, A k ( x, P )) dV k ≤ k . Recall that | V k | = 1 so, from Banach-Alaoglu and Riesz Theorems, there exists a varifold V such that,up to subsequences, V k → V in varifold sense (i.e weak convergence of Radon measures on G m ( N )). Ofcourse | V | = lim k | V k | = 1.Using the notation of [9] (see the Section 2) we have that the measure-function pairs ( V k , A k ) over G m ( N ), up to subsequences, satisfy the assumptions of Theorem 2.4. From (i) of the mentioned Theorem2.4, it follows that there exists a measure-function pair ( V, ˜ A ) with values in R S (i.e a Radon measure V on G m ( N ) and a matrix valued function ˜ A ∈ L loc ( V ) ) such that ( V k , A k ) ⇀ ( V, ˜ A ) (i.e V k ⌊ A k → V ⌊ ˜ A weak convergence of Radon vector valued measures).From Remark 2.6 we can express the generalized curvatures B k of the varifolds V k in terms of thesecond fundamental forms A k . Moreover, calling B the corresponding quantity to ˜ A , from the explicitexpression (8) it is clear that the weak convergence ( V k , A k ) ⇀ ( V, ˜ A ) implies the weak convergence( V k , B k ) ⇀ ( V, B ).Passing to the limit in k in (6) we see that ( V, B ) satisfies the equation, so V is an m -varifold withgeneralized curvature B .Now let us check that the corresponding generalized second fundamental form (in sense of equation(7)) to B is ˜ A .Call Λ lij ( x, P ) := P pj B ilp ( x, P ) − P pj P iq ∂Q lp ∂x q ( x )the corresponding second fundamental form to B and Λ k = A k the corresponding to B k .Since ( V k , B k ) ⇀ ( V, B ), from the definitions it is clear that ( V k , Λ k ) ⇀ ( V, Λ); but, from the definitionof ˜ A , ( V k , Λ k ) = ( V k , A k ) ⇀ ( V, ˜ A ). It follows that Λ = ˜ A V -almost everywhere and that ˜ A is thegeneralized second fundamental form of V associated to B .Finally, the lower semicontinuity of the functional (sentence (ii) of Theorem 2.4) implies Z G m ( N ) F ( x, P, ˜ A ) dV ≤ lim inf k Z G m ( N ) F ( x, P, A k ) dV k = 0 . From the assumption ii) of condition (2.2) on F it follows that ˜ A = 0 V -almost everywhere; henceforthwe constructed a non null m -varifold V in N with null second fundamental form and this concludes theproof. Remark 4.2.
A trivial but fundamental example of F : G m ( N ) × R S → R satisfying the assumptionsof Theorem 4.1 is F ( x, P, A ) = | A | p for any p > . Hence the Theorem implies that if a compact subset N of a Riemannian n -dimensional manifold ( ¯ N , g ) does not contain any non null k -varifold ( k ≤ n − )with null generalized second fundamental form then for every p > there exists a constant C p > suchthat | V | ≤ C p Z G m ( N ) | A | p dV for every k -varifold V in N with generalized second fundamental form A . Theorem 4.3.
Let N ⊂⊂ ¯ N be a compact subset of a (maybe non compact) n -dimensional Riemannianmanifold ( ¯ N , g ) (which, by Nash Embedding Theorem we can assume isometrically embedded in some R S ), fix m ≤ n − and let F : G m ( N ) × R S → R + be a function satisfying (2.2) .Assume that, for some m ≤ n − , the space ( N, g ) does not contain any non zero m -varifold withnull generalized second fundamental form.Consider a sequence { V k } k ∈ N ⊂ CV m ( N ) of curvature varifolds with generalized second fundamentalforms { A k } k ∈ N such that Z G m ( N ) F ( x, P, A k ) dV k ≤ C for some C > independent on k .Then there exists V ∈ CV m ( N ) with generalized second fundamental form A such that, up to subse-quences,i) ( V k , A k ) ⇀ ( V, A ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, A ) dV ≤ lim inf k R G m ( N ) F ( x, P, A k ) dV k . Proof . From Theorem 4.1 there exists a constant C F > F such that | V k | ≤ C F R G m ( N ) F ( x, P, A k ( x, P )) dV k , thus from the boundness of R G m ( N ) F ( x, P, A k ) dV k we have theuniform mass bound(29) | V k | ≤ C for some C > k . This mass bound, together with Banach Alaoglu and Riesz Theorems,implies that there exists an m -varifold V on N such that, up to subsequences, V k → V in varifold sense.In order to apply Hutchinson compactness Theorem 2.8 we have to prove that V actually is an integral m -varifold.From assumption iv ) on F of Definition 2.2, there exists a continuous function φ : G m ( N ) × [0 , ∞ ) → [0 , ∞ ), with 0 ≤ φ ( x, P, s ) ≤ φ ( x, P, t ) for 0 ≤ s ≤ t and ( x, P ) ∈ G m ( N ), φ ( x, P, t ) → ∞ locallyuniformly in ( x, P ) as t → ∞ , such that(30) φ ( x, P, | A | ) | A | ≤ F ( x, P, A )for all ( x, P, A ) ∈ G m ( N ) × R S . Since N is compact, also G m ( N ) is so and from the properties of φ there exists C > φ ( x, P, | A | ) ≥ | A | > C and any ( x, P ) ∈ G m ( N ). Thus for every k wecan split the computation of the L ( V k ) norm of A k as Z G m ( N ) | A k | dV k = Z G m ( N ) ∩{| A k |≤ C } | A k | dV k + Z G m ( N ) ∩{| A k | >C } | A k | dV k . The first term is bounded above by the mass bound (29). About the second term observe that, for | A | > C the inequality (30) implies that | A | ≤ F ( x, P, A ); then also the second term is bounded in virtueon the assumption that R G m ( N ) F ( x, P, A k ) dV k is uniformly bounded.We have proved that there exists a constant C such that, for all k ∈ N ,(31) Z G m ( N ) | A k | dV k ≤ C. Now, change point of view and look at the varifolds V k as curvature varifolds in R S . Recall (see Remark2.6) that the curvature function B can be written in terms of the generalized second fundamental form A relative to ¯ N and of the extrinsic curvature of the manifold ¯ N (as submanifold of R S ) which is uniformlybounded on N from the compactness assumption. Using the triangle inequality together with estimate1931) and the mass bound (29) we obtain the uniform estimate of the L ( V k ) norms of the curvaturefunctions B k (32) Z G m ( R S ) | B k | dV k ≤ C for some C > k .Estimate (32) and Remark 2.11 tell us that the integral varifolds V k of R S have uniformly boundedfirst variation: there exists a C > k such that | δV k ( X ) | ≤ C sup R S | X | , ∀ X ∈ C c ( R S ) vector field . The uniform bound on the first variations and on the masses of the integral varifolds V k allow us to applyAllard’s integral compactness Theorem (see for example [28] Remark 42.8 or the original paper of Allard[1]) and say that the limit varifold V is actually integral.The conclusions of the Theorem then follow from Hutchinson Theorem 2.8. Corollary 4.4.
Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of a (maybenon compact) n -dimensional Riemannian manifold ( ¯ N , g ) (which, by Nash Embedding Theorem can beassumed isometrically embedded in some R S ) and let F : G m ( N ) × R S → R + be a function satisfying (2.2) .Assume that, for some m ≤ n − , the space ( N, g ) does not contain any non zero m -varifold withnull generalized second fundamental form.Call (33) α mN,F := inf (Z G m ( N ) F ( x, P, A ) dV : V ∈ CV m ( N ) , V = 0 with generalized second fundamental form A ) and consider a minimizing sequence { V k } k ∈ N ⊂ CV m ( N ) of curvature varifolds with generalized secondfundamental forms { A k } k ∈ N such that Z G m ( N ) F ( x, P, A k ) dV k ↓ α mN,F . Then there exists V ∈ CV m ( N ) with generalized second fundamental form A such that, up to subse-quences,i) ( V k , A k ) ⇀ ( V, A ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, A ) dV ≤ α mF . Proof . We only have to check that α mN,F < ∞ , then the conclusion follows from Theorem 4.3. Butthe fact is trivial since int ( N ) = ∅ , indeed we can always construct a smooth compact m -dimensionalembedded submanifold of N , which of course is a curvature m -varifold with finite energy. Remark 4.5.
Notice that, a priori, Corollary 4.4 does not ensure the existence of a minimizer since itcan happen that the limit m -varifold V is null. In the next Section 5 we will see that, if F ( x, P, A ) ≥ C | A | p for some C > and p > m , then this is not the case and we have a non trivial minimizer. In this Subsection we adapt to the context of varifolds with weak mean curvature the results of the previousSubsection 4.1 about varifolds with generalized second fundamental form (for the basic definitions andproperties see Section 2). The following Isoperimetric Inequality involving the weak mean curvature canbe seen as a variant of Theorem 2.3 in [31]. 20 heorem 4.6.
Let N ⊂⊂ ¯ N be a compact subset of a (maybe non compact) n -dimensional Riemannianmanifold ( ¯ N , g ) (which, by Nash Embedding Theorem we can assume isometrically embedded in some R S )and let m ≤ n − . Then the following conditions are equivalent:i) N contains no nonzero m -varifold with null weak mean curvature relative to ¯ N (i.e N contains nononzero stationary m -varifold; see Remark 2.14).ii) There is an increasing function Φ : R + → R + with Φ(0) = 0 and a function F : G m ( N ) × R S → R + satisfying (2.2) (see Section 2) such that for every m -varifold V in N with weak mean curvature H N relative to ¯ N | V | ≤ Φ (cid:16) Z G m ( N ) F ( x, P, H N ( x, P )) dV (cid:17) . iii) for every function F : G m ( N ) × R S → R + satisfying (2.2) (see Section 2) there exists a constant C F > such that for every m -varifold V in N with weak mean curvature H N relative to ¯ N | V | ≤ C F Z G m ( N ) F ( x, P, H N ( x, P )) dV. Proof . The proof is similar to the proof of Theorem 4.1. Of course iii) ⇒ ii) ⇒ i). We prove bycontradiction that i) ⇒ iii): assume that iii) is not satisfied and show that also i) cannot be satisfied.First fix the function F . If iii) is not satisfied then there exists a sequence { V k } k ∈ N of m -varifolds in N with weak mean curvatures H Nk relative to ¯ N (see Definition 2.12) such that | V k | ≥ k Z G m ( N ) F ( x, P, H Nk ( x, P )) dV k . We can assume that | V k | = 1 otherwise replace V k with the normalized varifold ˜ V k := | V k | V k (observe thatthe weak mean curvature is invariant under this rescaling of the measure and that R G m ( N ) F ( x, P, H Nk ) dV k = | V k | R G m ( N ) F ( x, P, H Nk ) d ˜ V k ). Hence Z G m ( N ) F ( x, P, H Nk ( x, P )) dV k ≤ k . Recall that | V k | = 1 so, from Banach-Alaoglu and Riesz Theorems, there exists a varifold V such that,up to subsequences, V k → V in varifold sense (i.e weak convergence of Radon measures on G m ( N )). Ofcourse | V | = lim k | V k | = 1.Now the measure-function pairs ( V k , H Nk ) over G m ( N ), up to subsequences, satisfy the assumptionsof Theorem 2.4 and (i) (of the mentioned Theorem 2.4) implies that there exists a measure-function pair( V, ˜ H N ) with values in R S such that ( V k , H Nk ) ⇀ ( V, ˜ H N ) weak convergence of measure-function pairs(see Definition 2.1).At this point we have to check that V is an m -varifold of N with weak mean curvature ˜ H N relativeto ¯ N . Recall that N ֒ → R S , so the varifolds V k can be seen as varifolds with weak mean curvatures H R S k in R S ; from equation (11), the measure-function pair convergence ( V k , H Nk ) ⇀ ( V, ˜ H N ) impliesthe measure-function pair convergence ( V k , H R S k ) ⇀ ( V, ˜ H N + P jk ∂Q ij ∂x k ) which says ( pass to the limitin Definition 2.10) that V is an m -varifold in R S with weak mean curvature ˜ H N + P jk ∂Q ij ∂x k . Thus, byDefinition 2.12, V is an m -varifold of N with weak mean curvature H N := ˜ H N relative to ¯ N .Finally, the lower semicontinuity of the functional (sentence (ii) of Theorem 2.4) implies Z G m ( N ) F ( x, P, H N ) dV ≤ lim inf k Z G m ( N ) F ( x, P, H Nk ) dV k = 0 . From the assumption ii) of condition (2.2) on F it follows that H N = 0 V -almost everywhere; henceforthwe constructed a non null m -varifold V in N with null weak mean curvature relative to ¯ N and thisconcludes the proof.We also have a counterpart of Theorem 4.3 concerning the weak mean curvature:21 heorem 4.7. Let N ⊂⊂ ¯ N be a compact subset of a (maybe non compact) n -dimensional Riemannianmanifold ( ¯ N , g ) (which, by Nash Embedding Theorem we can assume isometrically embedded in some R S ), fix m ≤ n − and let F : G m ( N ) × R S → R + be a function satisfying (2.2) .Assume that, for some m ≤ n − , the space ( N, g ) does not contain any non zero m -varifold withnull weak mean curvature relative to ¯ N .Consider a sequence { V k } k ∈ N ⊂ HV m ( N ) of integral m -varifolds with weak mean curvatures { H Nk } k ∈ N relative to ¯ N such that Z G m ( N ) F ( x, P, H Nk ) dV k ≤ C for some C > independent on k .Then there exists V ∈ HV m ( N ) integral varifold with weak mean curvature H N relative to ¯ N suchthat, up to subsequences,i) ( V k , H Nk ) ⇀ ( V, H N ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, H N ) dV ≤ lim inf k R G m ( N ) F ( x, P, H Nk ) dV k . Proof . The proof is analogous to the proof of Theorem 4.3. From Theorem 4.6 there exists a constant C F > F such that | V k | ≤ C F R G m ( N ) F ( x, P, H Nk ( x, P )) dV k , thus from theboudness of R G m ( N ) F ( x, P, H Nk ) dV k we have the uniform mass bound(34) | V k | ≤ C for some C > k . This mass bound, together with Banach Alaoglu and Riesz Theorems,implies that there exists an m -varifold V on N such that, up to subsequences, V k → V in varifold sense.The proof that V actually is an integral m -varifold is completely analogous to the same statement inthe proof of Theorem 4.3: formally substituting H Nk to A k in the mentioned proof we arrive to(35) Z G m ( N ) | H Nk | dV k ≤ C. Now, change point of view and look at the varifolds V k as integral varifolds in R S . From Definition 2.12the weak mean curvature H R S k in R S can be written in terms of H Nk and of the extrinsic curvature of themanifold ¯ N (as submanifold of R S ) which is uniformly bounded on N from the compactness assumption.Using the triangle inequality together with estimate (35) and the mass bound (34) we obtain the uniformestimate of the L ( V k ) norms of the weak mean curvatures H R S k (36) Z G m ( R S ) | H R S k | dV k ≤ C for some C > k . It follows (see Definition 2.10) that the integral varifolds V k of R S have uniformly bounded first variation: there exists a constant C > k such that | δV k ( X ) | ≤ C sup R S | X | , ∀ X ∈ C c ( R S ) vector field . The uniform bound on the first variations and on the masses of the integral varifolds V k allow us to applyAllard’s integral compactness Theorem (see for example [28] Remark 42.8 or the original paper of Allard[1]) and say that the limit varifold V is actually integral.With the same arguments in the end of the proof of Theorem 4.6, one can show that the varifoldconvergence of a subsequence V k → V and the uniform energy bound R G m ( N ) F ( x, P, H Nk ) dV k < C impliesthe existence of a measure-function pair converging subsequence ( V k , H Nk ) ⇀ ( V, H N ) for some R S -valuedfunction H N ∈ L loc ( V ) which actually is the weak mean curvature of V relative to ¯ N .We conclude that V ∈ HV m ( N ) is an integral m -varifold of N with weak mean curvature H N relative to ¯ N and i) holds; property ii) follows from the general Theorem 2.8 about measure-function pairconvergence (specifically see sentence ii) of the mentioned Theorem).Finally we have a counterpart of Corollary 4.4 22 orollary 4.8. Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of a (maybenon compact) n -dimensional Riemannian manifold ( ¯ N , g ) (which, by Nash Embedding Theorem can beassumed isometrically embedded in some R S ) and let F : G m ( N ) × R S → R + be a function satisfying (2.2) .Assume that, for some m ≤ n − , the space ( N, g ) does not contain any non zero m -varifold withnull weak mean curvature relative to ¯ N .Call (37) β mN,F := inf (Z G m ( N ) F ( x, P, H N ) dV : V ∈ HV m ( N ) , V = 0 with weak wean curvature H N relative to ¯ N ) and consider a minimizing sequence { V k } k ∈ N ⊂ HV m ( N ) of integral varifolds with weak mean curvatures { H Nk } k ∈ N such that Z G m ( N ) F ( x, P, H Nk ) dV k ↓ β mN,F . Then there exists an integral m -varifold V ∈ HV m ( N ) with weak mean curvature H N relative to ¯ N such that, up to subsequences,i) ( V k , H Nk ) ⇀ ( V, H N ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, H N ) dV ≤ β mN,F . Proof . As in Corollary 4.4 we have that β mN,F < ∞ , then the conclusion follows from Theorem 4.7. Remark 4.9.
As for the generalized second fundamental form, a priori, Corollary 4.4 does not ensurethe existence of a minimizer since it can happen that the limit m -varifold V is null. In Section 6 we willsee that, if F ( x, P, H N ) ≥ C | H N | p for some C > and p > m , then this is not the case and we have anon trivial minimizer. F ( x, P, A ) ≥ C | A | p with p > m : non degeneracy of theminimizing sequence and existence of a C ,α minimizer Throughout this Section, ( ¯
N , g ) stands for a compact n -dimensional Riemannian manifold isometricallyembedded in some R S (by Nash Embedding Theorem) and N ⊂⊂ ¯ N is a compact subset with non emptyinterior (as subset of N ). Fix m ≤ n −
1; we will focus our attention and specialize the previous techniquesto the case F : G m ( N ) × R S → R + is a function satisfying (2.2) F ( x, P, A ) ≥ C | A | p for some p > m and C > . (38)Recall that we are considering the minimization problem α mN,F := inf (Z G m ( N ) F ( x, P, A ) dV : V ∈ CV m ( N ) , V = 0 with generalized second fundamental form A ) . Our goal is to prove the existence of a minimizer for α mN,F , F as in (38).Let { V k } k ∈ N ⊂ CV m ( N ) be a minimizing sequence of curvature varifolds with generalized secondfundamental forms { A k } k ∈ N such that Z G m ( N ) F ( x, P, A k ) dV k ↓ α mN,F ;from Corollary 4.4 we already know that there exists V ∈ CV m ( N ) with generalized second fundamentalform A such that, up to subsequences, 23) ( V k , A k ) ⇀ ( V, A ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, A ) dV ≤ α mN,F .In order to have the existence of a minimizer we only have to check that V is not the zero varifold;this will be done in the next Subsection 5.1 using the estimates of Section 3. First of all, since N ⊂ R S , a curvature m -varifold V of N can be seen as a curvature varifold in R S (for theprecise value of the curvature function B in R S see Remark 2.6); as before we write V = V ( M, θ ) where M is a rectifiable set and θ is the integer multiplicity function. Let us call H R s the weak mean curvatureof V as integral m -varifold in R S and, as in Section 3, let us denote with µ = µ V = H m ⌊ θ = π ♯ V thespatial measure associated to V and with spt µ its support. Lemma 5.1.
Let N ⊂⊂ ¯ N be a compact subset of the n -dimensional Riemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding Theorem) and fix p > . Then there exists aconstant C N,p > depending only on p and N such that for every V = V ( M, θ ) ∈ CV m ( N ) curvature m -varifold of N Z M (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) p dµ ≤ C N,p | V | + Z G m ( N ) | A | p dV ! . Proof . Recall (see Remark 2.6) that it is possible to write the curvature function B of V seen ascurvature m -varifold of R S in terms of the second fundamental form A relative to ¯ N and the curvatureof the manifold ¯ N seen as submanifold of R S (the terms involving derivatives of Q ): B ijk = A kij + A jik + P jl P iq ∂Q lk ∂x q ( x ) + P kl P iq ∂Q lj ∂x q ( x ) . From Remark 2.11 the weak mean curvature H R S , which is a vector of R S , can be written in terms of B as (cid:16) H R S (cid:17) i = S X j =1 B jij = S X j =1 (cid:18) A jji + A ijj + P il P jq ∂Q lj ∂x q ( x ) + P jl P jq ∂Q li ∂x q ( x ) (cid:19) i = 1 . . . , S. Notice that, since N ⊂⊂ ¯ N is a compact subset of the manifold ¯ N smoothly embedded in R S , the functions ∂Q lj ∂x m are uniformly bounded by a constant C N depending on the embedding N ֒ → R S ; moreover the P jm are projection matrices so they are also uniformly bounded and we can say that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X j,l,m =1 P il P jq ∂Q lj ∂x q + P jl P jq ∂Q li ∂x q i =1 ,...,S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N as vector of R S .About the first term observe that, from the triangle inequality applied to the R S -vectors ( A jji ) i =1 ,...,S ( j is fixed for each single vector), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X j =1 A jji i =1 ,...,S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ S X j =1 (cid:12)(cid:12)(cid:12) ( A jji ) i =1 ,...,S (cid:12)(cid:12)(cid:12) ≤ S | A | where, of course | A | := qP Si,j,k =1 ( A ijk ) ≥ | ( A jji ) i =1 ,...,S | for all j = 1 , . . . , S . The second adding term isanalogous.Putting together the two last estimates, by a triangle inequality, we have (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) ≤ S | A | + C N . a + b ) p ≤ p − ( a p + b p ) for a, b ≥ p > t t p for t ≥ , p > (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) p ≤ C N,p ( | A | p + 1) . With an integration we get the conclusion.Using the estimates of Section 3 and the last Lemma we have uniform lower bounds on the massand on the diameter of the spatial support of a curvature m -varifold V ∈ CV m ( N ) of N with bounded R G m ( N ) | A | p dV , p > m . Theorem 5.2.
Let N ⊂⊂ ¯ N be a compact subset of the n -dimensional Riemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding Theorem) and fix m ≤ n − , p > m .Then there exists a constant C N,p,m > depending only on p, m and on the embedding of N into R S such that C N,p,m ↑ + ∞ as p ↓ m and such that for every V = V ( M, θ ) ∈ CV m ( N ) curvature m -varifoldof N with spatial measure µi ) diam ¯ N (spt µ ) ≥ C N,p,m (cid:16) | V | + R G m ( N ) | A | p dV (cid:17) p − m (40) where diam ¯ N (spt µ ) is the diameter of spt µ as a subset of the Riemannian manifold ¯ N ; ii ) C N,p,m | V | | V | + Z G m ( N ) | A | p dV ! mp − m ≥ . (41) Notice that ii ) implies the existence of a constant a N,m,p, R | A | p > depending only on p, m , on R G m ( N ) | A | p dV and on the embedding of N into R S , with a N,p,m, R | A | p ↓ if p ↓ m or if R G m ( N ) | A | p dV ↑ + ∞ such that | V | ≥ a N,p,m, R | A | p > . Proof . i ) From Lemma 3.7 diam ¯ N (spt µ ) ≥ diam R S (spt µ ) ≥ C p,m (cid:16) R M | H | p dµ (cid:17) p − m where C p,m > p, m and such that C p,m → ∞ if p ↓ m . Theconclusion follows plugging into the last inequality the estimate of Lemma 5.1. ii ) From Lemma 3.8, | V | ≥ C p,m (cid:16) R M | H | p dµ (cid:17) mp − m with C p,m > Corollary 5.3.
Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of the n -dimensional Riemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding The-orem) and fix m ≤ n − .Assume that the space ( N, g ) does not contain any non zero m -varifold with null generalized secondfundamental form and consider a function F : G m ( N ) × R S → R + satisfying (2.2) , (38) and a cor-responding minimizing sequence of curvature m -varifolds { V k } k ∈ N ⊂ CV m ( N ) with generalized secondfundamental forms { A k } k ∈ N such that Z G m ( N ) F ( x, P, A k ) dV k ↓ α mN,F for the definition of α mN,F see (33) ). Then, called µ k the spatial measures associated to V k , there existsa constant a N,F,m > depending only on N , F and m such that i ) diam ¯ N (spt µ k ) ≥ a N,F,m (42) ii ) | V k | ≥ a N,F,m . (43) Proof . From Theorem 4.1 and the finiteness of α mN,F , since ( N, g ) does not contain any non zero m -varifold with null generalized second fundamental form, | V k | ≤ C N,F,m Z G m ( N ) F ( x, P, A k ) dV k ≤ C N,F,m for some C N,F,m > N, F and m .Moreover, since (by assumption (38)) F ( x, P, A ) ≥ C | A | p for some p > m and C >
0, the boundness of R G m ( N ) F ( x, P, A k ) dV k implies that Z G m ( N ) | A k | p dV k ≤ C N,F,m for some C N,F,m>
N, F and m .The conclusion follows putting the last two inequalities into Theorem 5.2. Collecting Corollary 4.4 and Corollary 5.3 we can finally state and prove the first main Theorem 1.3.
Proof of Theorem 1.3 If a ) is true we are done, so we can assume that a ) is not satisfied.Let { V k } k ∈ N ⊂ CV m ( N ) with generalized second fundamental forms { A k } k ∈ N be a minimizing se-quence of α mN,F : Z G m ( N ) F ( x, P, A k ) dV k ↓ α mN,F . Called µ k the spatial measures associated to V k , from Corollary 5.3 we have the lower bounds: i ) diam ¯ N (spt µ k ) ≥ a N,F,m ii ) | V k | ≥ a N,F,m , for a constant a N,F,m > N , F and m . Corollary 4.4 implies the existence of acurvature m -varifold V = V ( M, θ ) ∈ CV m ( N ) with generalized second fundamental form A such that,up to subsequences,i) ( V k , A k ) ⇀ ( V, A ) in the weak sense of measure-function pairs of N ,ii) R G m ( N ) F ( x, P, A ) dV ≤ α mN,F .The measure-function pair convergence implies the varifold convergence of V k → V and the convergenceof the associated spatial measures π ♯ V k =: µ k → µ := π ♯ V weak convergence of Radon measures on N .It follows that 0 < a N,F,m ≤ | V k | = µ k ( N ) → µ ( N ) = | V | , thus V = 0 is a minimizer for α mN,F .Proof of b ): V is indecomposable in CV m ( N ). Otherwise V = V + V with 0 = V , V ∈ CV m ( N ); theminimizing property of V implies that, up to exchanging V with V , we must have R G m ( N ) F ( x, P, A ) dV =26 G m ( N ) F ( x, P, A ) dV = α mN,F and V is a non zero m -varifold with null generalized second fundamentalform, contradicting the assumption that a ) does not hold.Proof of b ): since N ֒ → R S is properly embedded, the weak convergence µ k → µ on N implies theweak convergence of µ k → µ as Radon measures on R S . From mass bound on the V k and the bound on R G m ( N ) | A k | p dV k given by the assumption (38) on F , Lemma 5.1 allows us to apply Proposition 3.9 andwe can say that the spatial supportsspt µ k → spt µ Hausdorff convergence as subsets of R S . Notice that, since ¯
N ֒ → R S is embedded, the Hausdorff convergence of M k → M as subsets of R S impliesspt µ k → spt µ Hausdorff convergence as subsets of ¯
N , and this implies that 0 < a
N,F,m ≤ lim k diam ¯ N (spt µ k ) = diam ¯ N (spt µ ) . hence b V ∈ CV ( N ) is a non null curvature varifold on N with generalized secondfundamental form A (relative to ¯ N ) in L p ( V ) for some p > m . Since N ֒ → R S , V can also be seen asa varifold in R S and Remark 2.6 tell that V is actually a varifold with generalized curvature function B given by B ijk = A kij + A jik + P jl P iq ∂Q lk ∂x q ( x ) + P kl P iq ∂Q lj ∂x q ( x )where the terms of the type P jl P iq ∂Q lk ∂x q ( x ) represent the extrinsic curvature of ¯ N as a submanifold of R S and, of course, are bounded on N from the compactness:sup x ∈ N (cid:12)(cid:12)(cid:12)(cid:12) P jl P iq ∂Q lk ∂x q ( x ) + P kl P iq ∂Q lj ∂x q ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C N . Hence, from triangle inequality, | B | ≤ | A | + C N and | B | p ≤ C N,p ( | A | p + 1) . Using the mass bound | V | = lim k | V k | ≤ C < ∞ , with an integration we get Z G m ( R S ) | B | p dV < ∞ . Under this conditions Hutchinson shows in [10] that V is a locally a graph of multivalued C ,α functionsand that b
3) holds. m -varifold with weak mean curvatureminimizing R | H | p for p > m As before, throughout this Section ( ¯
N , g ) stands for a compact n -dimensional Riemannian manifoldisometrically embedded in some R S (by Nash Embedding Theorem) and N ⊂⊂ ¯ N is a compact subsetwith non empty interior (as subset of N ). Fix m ≤ n −
1; analogously to Section 5 we will focus ourattention to the case F : G m ( N ) × R S → R + is a function satisfying (2.2) F ( x, P, H ) ≥ C | H | p for some p > m and C > . (44) 27ecall that we are considering the minimization problem β mN,F := inf (Z G m ( N ) F ( x, P, H N ) dV : V ∈ HV m ( N ) , V = 0 with weak mean curvature H N relative to ¯ N ) . Our goal is to prove the existence of a minimizer for β mN,F , F as in (44).As in Section 5 we consider a minimizing sequence { V k } k ∈ N ⊂ HV m ( N ) of integral m -varifolds withweak mean curvatures { H Nk } k ∈ N relative to ¯ N such that Z G m ( N ) F ( x, P, H Nk ) dV k ↓ β mN,F ;from Corollary 4.8 we already know that there exists V ∈ HV m ( N ) with with weak mean curvature H N relative to ¯ N such that, up to subsequences,i) ( V k , H Nk ) ⇀ ( V, H N ) in the weak sense of measure-function pairs,ii) R G m ( N ) F ( x, P, H N ) dV ≤ β mN,F .In order to have the existence of a minimizer we only have to check that V is not the zero varifold;this will be done analogously to Subsection 5.1 using the estimates of Section 3.As before, since N ⊂ R S , an integral m -varifold V of N with weak mean curvature H N relative to ¯ N can be seen as integral m -varifold of R S with weak mean curvature H R S . We write V = V ( M, θ ) where M is a rectifiable set and θ is the integer multiplicity function; finally, as in Section 3, let us denote with µ = µ V = H m ⌊ θ = π ♯ V the spatial measure associated to V and with spt µ the spatial support of V . Lemma 6.1.
Let N ⊂⊂ ¯ N be a compact subset of the n -dimensional Riemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding Theorem) and fix p > . Then there exists aconstant C N,p > depending only on p and N such that for every V = V ( M, θ ) ∈ HV m ( N ) integral m -varifold of N with weak mean curvature H N relative to ¯ N Z M (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) p dµ ≤ C N,p | V | + Z G m ( N ) | H N | p dV ! . Proof . By Definition 2.12 we can express( H R S ) i = ( H N ) i + P jk ∂Q ij ∂x k and from the triangle inequality(45) (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) H N (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) P jk ∂Q ij ∂x k (cid:12)(cid:12)(cid:12)(cid:12) ;as vectors in R S . The second summand of the right hand side is a smooth function on the compact set G m ( N ) hence bounded by a constant C N depending on N : (cid:12)(cid:12)(cid:12)(cid:12) P jk ∂Q ij ∂x k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C N . Using the standard inequality ( a + b ) p ≤ p − ( a p + b p ) for a, b ≥ p > (cid:12)(cid:12)(cid:12) H R S (cid:12)(cid:12)(cid:12) p ≤ C N,p (cid:0) (cid:12)(cid:12) H N (cid:12)(cid:12) p (cid:1) which gives the thesis with an integration. Remark 6.2.
An analogous result to Theorem 5.2 holds, just replace V = V ( M, θ ) ∈ CV m ( N ) with V = V ( M, θ ) ∈ HV m ( N ) and R G m ( N ) | A | p dV with R G m ( N ) | H N | p dV . β mN,F , F as in 2.2, (44). Lemma 6.3.
Let N ⊂⊂ ¯ N be a compact subset with non empty interior, int ( N ) = ∅ , of the n -dimensionalRiemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding Theorem) and fix m ≤ n − .Assume that the space ( N, g ) does not contain any non zero m -varifold with null weak mean curvature H N relative to ¯ N and consider a function F : G m ( N ) × R S → R + satisfying (2.2) , (44) and a corre-sponding minimizing sequence of integral m -varifolds { V k } k ∈ N ⊂ HV m ( N ) with weak mean curvatures { H Nk } k ∈ N relative to ¯ N such that Z G m ( N ) F ( x, P, H Nk ) dV k ↓ β mN,F ( for the definition of β mN,F see (37) ). Then, called µ k the spatial measures of V k , there exists a constant b N,F,m > depending only on N , F and m such that i ) diam ¯ N (spt µ k ) ≥ b N,F,m (46) ii ) | V k | ≥ b N,F,m . (47) Proof . From Theorem 4.6 and the finiteness of β mN,F , since ( N, g ) does not contain any non zero m -varifold with null weak mean curvature H N relative to ¯ N , | V k | ≤ C N,F,m Z G m ( N ) F ( x, P, H Nk ) dV k ≤ C N,F,m for some C N,F,m > N, F and m .Moreover, since (by assumption (44)) F ( x, P, H N ) ≥ C | H N | p for some p > m and C >
0, the boundnessof R G m ( N ) F ( x, P, H Nk ) dV k implies that Z G m ( N ) | H Nk | p dV k ≤ C N,F,m for some C N,F,m>
N, F and m .The conclusion follows from the last two inequalities and Remark 6.2.Now, collecting Corollary 4.8 and Lemma 6.3 we can finally state and prove Theorem 1.1, namely theexistence of a non trivial minimizer for β mN,F , F as in 2.2, (44). Proof of Theorem 1.1 If a ) is true we are done, so we can assume that a ) is not satisfied.Let { V k } k ∈ N ⊂ HV m ( N ) with weak mean curvatures { H Nk } k ∈ N be a minimizing sequence of β mN,F : Z G m ( N ) F ( x, P, H Nk ) dV k ↓ β mN,F . Called µ k the spatial measures of V k , from Lemma 6.3 we have the lower bounds: i ) diam ¯ N (spt µ k ) ≥ b N,F,m > ii ) | V k | ≥ b N,F,m > , for a constant b N,F,m > N , F and m . Corollary 4.8 implies the existence of an integral m -varifold V ∈ HV m ( N ) with weak mean curvature H N relative to ¯ N such that, up to subsequences,i) ( V k , H Nk ) ⇀ ( V, H N ) in the weak sense of measure-function pairs of N ,ii) R G m ( N ) F ( x, P, H N ) dV ≤ β mN,F . 29nalogously to the proof of Theorem 1.3, one shows that0 < b N,F,m ≤ | V k | = µ k ( N ) → µ ( N ) = | V | , thus V = 0 is a minimizer for β mN,F . The proof of b
1) and b
2) are again analogous to the proof of thecorresponding sentences in Theorem 1.3. Let us just comment on b V k and the bound on R G m ( N ) | H Nk | p dV k given by the assumption (44) on F , Lemma 6.1 allows us to applyProposition 3.9 and, using the same tricks of Theorem 1.3 we can say that the spatial supportsspt µ k → spt µ Hausdorff convergence as subsets of ¯
N , and this implies that 0 < b
N,F,m ≤ lim k diam ¯ N (spt µ k ) = diam ¯ N (spt µ ) , hence b First of all let us point out that our setting includes, speaking about ambient manifolds, a large class ofRiemannian manifolds with boundary.
Remark 7.1.
Notice that if N is a compact n -dimensional manifold with boundary then there exists an n -dimensional (a priori non compact) manifold ¯ N without boundary such that N is a compact subset of ¯ N (to define ¯ N just extend N a little beyond ∂N in the local boundary charts). Hence, given a compact n -dimensional Riemannian manifold ( N, g ) with boundary such that the metric g can be extended in asmooth and non degenerate way (i.e. g positive definite) up to the boundary ∂N , then N is isometric toa compact subset of an n -dimensional Riemannian manifold ( ¯ N , ¯ g ) without boundary.Thus all the Lemmas, Propositions and Theorems apply to the case in which the ambient space is aRiemannian manifold with boundary with the described non degeneracy property at ∂N . Now let us show that the main results Theorem 1.3 and Theorem 1.1 are non empty, i.e we haveexamples of compact subsets of Riemannian manifolds where do not exist non zero varifolds with nullweak mean curvature relative to ¯ N and a fortiori there exists no non zero varifold with null generalizedsecond fundamental form. Let us start with an easy Lemma: Lemma 7.2.
Let N ⊂⊂ ¯ N be a compact subset of the n -dimensional Riemannian manifold ( N, g ) isometrically embedded in some R S (by Nash Embedding Theorem), fix m ≤ n − and assume that N contains no non zero m -varifold with null weak mean curvature relative to ¯ N . Then N does not containany non zero m -varifold with null generalized second fundamental form. Proof . We show that if the varifold V has null generalized second fundamental form relative to ¯ N then V also has null weak mean curvature relative to ¯ N . Indeed let V be a varifold on N with generalizedcurvature function B and second fundamental form A relative to ¯ N , then, from Remark 2.6, B ijk = A kij + A jik + P jl P ip ∂Q lk ∂x p ( x ) + P kl P ip ∂Q lj ∂x p ( x )where P and Q ( x ) are the projection matrices on P ∈ G m ( N ) and T x ¯ N . Moreover, from Remark 2.11, V has weak mean curvature as a varifold in R S ( H R S ) i = B jij ;hence, if the generalized second fundamental form A is null, then( H R S ) i = P il P jk ∂Q lj ∂x k ( x ) + P jl P jk ∂Q li ∂x k ( x ) .
30t is not hard to check that the first summand of the right hand side is null (fix a point x of ¯ N and choosea base of T x ¯ N in which the Christoffel symbols of ¯ N vanish at x ; write down the orthogonal projectionmatrix Q with respect to this base and check the condition in this frame). Thus H R S i = P jk ∂Q ij ∂x k andDefinition 2.12 gives ( H N ) i = ( H R S ) i − P jk ∂Q ij ∂x k ( x ) = 0 . Collecting Lemma 7.2 and Remark 2.13 we can say that if a compact subset N ⊂⊂ ¯ N has a nonzero m -varifold with null generalized second fundamental form, then a fortiori N contains a non zero m -varifold with null weak mean curvature relative to ¯ N , then a fortiori N contains a non zero m -varifoldwith null first variation relative to ¯ N (recall, see Remark 2.14, that a varifold with null first variation isalso called stationary varifold). Hence it is enough to give examples of compact subsets of Riemannianmanifolds which do not contain any non zero m -varifold with null first variation relative to ¯ N .First, we mention two examples given by White in [31] (for the proofs we refer to the original paper)next we will propose a couple of new examples which can be seen as a sort of generalization of White’sones. Recall that if N is a compact Riemannian manifold with smooth boundary, N is said to be meanconvex provided that the mean curvature vector at each point of ∂N is an nonnegative multiple of theinward-pointing unit normal. Example 7.3.
Suppose that N is a compact, connected, mean convex Riemannian manifold with smooth,nonempty boundary, and that no component of ∂N is a minimal surface. Suppose also that the dimension n of N is at most and that the Ricci curvature of N is everywhere positive. Then N contains no nonzero n − -varifold with null first variation relative to N (i.e. stationary n − -varifold).More generally, if N has nonnegative Ricci curvature, then the same conclusion holds unless N con-tains a closed, embedded, totally geodesic hypersurface M such that Ric ( ν, ν ) = 0 for every unit normal ν to M (where Ric is the Ricci tensor of N ). Minimal surfaces in ambient manifolds of the form M × R have been deeply studied in recent years(see for example [19], [20] and [25]); notice that M × R is foliated by the minimal surfaces M × { z } .In the second example we can see that very general compact subsets of ambient spaces admitting suchfoliations do not contain non zero codimension 1 varifolds with null first variation. Example 7.4.
Let ¯ N be an n -dimensional Riemannian manifold. Let f : ¯ N → R be a smooth functionwith nowhere vanishing gradient such that the level sets of f are minimal hypersurfaces or, more generally,such that the sublevel sets { x : f ( x ) ≤ z } are mean convex. Let N be a compact subset of ¯ N such thatfor each z ∈ R , no connected component of f − ( z ) is a minimal hypersurface lying entirely in N . Then N contains no non zero n − -varifold with null first variation relative to ¯ N . Observe that both examples concern the non-existence of codimension 1 stationary varifolds: nextwe propose a couple of new examples in higher codimension. We need the following maximum principlefor stationary (i.e. with null first variation) varifolds given by White, for the proof see [32], Theorem 1.Before stating it recall that if N is an n -dimensional Riemannian manifold with boundary ∂N , N is said strongly m -convex at a point p ∈ ∂N provided k + k + . . . + k m > k ≤ k ≤ . . . ≤ k n − are the principal curvatures of ∂N at p with respect to the unit normal ν N that points into N . Theorem 7.5.
Let ¯ N be a smooth Riemannian manifold of dimension n , let N ⊂ ¯ N be a smoothRiemannian n -dimensional manifold with boundary, and assume p to be a point in ∂N at which N isstrongly m -convex. Then p is not contained in the support of any m -varifold in N with null first variationrelative to ¯ N . Theorem 7.6.
Let ¯ N be an n -dimensional Riemannian manifold and consider as ambient manifold ¯ N × R S , s > with the product metric. Then any compact subset N ⊂⊂ ¯ N × R S does not contain anynon null stationary n + k -varifold, k = 1 , . . . , s − (i.e. n + k -varifold with null first variation relative to ¯ N × R S ). Proof . Assume by contradiction that V is a non null n + k -varifold in N with null first variation in¯ N × R S for some 1 ≤ k ≤ s −
1. Consider the function ρ : ¯ N × R S → R + defined as¯ N × R S ∋ ( x, y ) ρ ( x, y ) := | y | R S where of course | y | R S is the norm of y as vector of R S . With abuse of notation, call M ⊂ N the spatialsupport of V (now M may not be rectifiable, it is just compact); observe that, since M is compact, thenthe function ρ restricted to M has a maximum r > x , y ) ∈ M ⊂ ¯ N × R S (observethat the maximum r has to be strictly positive otherwise we would have a non null n + k -varifold inan n -dimensional space, which clearly is not possible by the very definition of varifold). It follows that,called ¯ N r the tube of center ¯ N and radius r ¯ N r := { ( x, y ) : x ∈ ¯ N , | y | R S ≤ r } , the spatial support of V is contained in ¯ N r :(48) M ⊂ ¯ N r . Moreover M is tangent to the hypersurface C r := ∂ ¯ N r = { ( x, y ) : x ∈ ¯ N , | y | R S = r } at the point ( x , y ).Observe that C r is diffeomorphic to ¯ N × rS s − R S , where of course rS s − R S is the s − R S of radius r centered in the origin.Using normal coordinates in ¯ N × R S it is a simple exercise to observe that the principal curvaturesof C r with respect to the inward pointing unit normal are constantly k = k = . . . = k n = 0 , k n +1 = k n +2 = . . . = k s − = 1 r (just observe that the inward unit normal is − Θ, where Θ is the radial vector which parametrizes S s − R S ; ofcourse − Θ is constant respect to the x coordinates; using normal coordinates one checks that the secondfundamental form is made of two blocks: the one corresponding to ¯ N is null and the other one coincideswith the second fundamental form of S s − R S as hypersurface in R S ).It follows that C r = ∂ ¯ N r is strongly n + k -convex in all of its points, for all 1 ≤ k ≤ n −
1; but V is a non null n + k -varifold in ¯ N r with null first variation relative to ¯ N and tangent to C r at the point( x , y ) ∈ C r ∩ M . Fact which contradicts the maximum principle, Theorem 7.5.As a corollary we have an example in all the codimensions in R S : Theorem 7.7.
Let N ⊂⊂ R S be a compact subset of R S , s > . Then, for all ≤ m ≤ s − , N containsno non zero m -varifold with null first variation relative to R S . Proof . Just take ¯ N := { x } in the previous example, Theorem 7.6, and observe that { x } × R S isisometric to R S .Otherwise argue by contradiction as in the proof of Theorem 7.6 and observe that the support of thenon zero m -varifold with null first variation is contained in a ball of R S and tangent to its boundary,namely a sphere. Of course the sphere is strongly m -convex; it follows a contradiction with the maximumprinciple, Theorem 7.5. 32 emark 7.8. Recall that if the ambient n -dimensional Riemannian manifold N is compact withoutboundary, then Almgren proved in [2] that for every ≤ m < n there exists an integral m -varifold withnull first variation relative to N . Moreover, in the same setting of compact N and ∂N = ∅ , Schoen andSimon [26], using the work of Pitts [29], proved that N must contain a closed, embedded hypersurface withsingular set of dimension at most n − . Hence, the isoperimetric inequality Theorem 4.6 fails for such N and the Theorem 1.1 is trivially true. However, as written above, there are many interesting examplesof ambient manifolds with boundary where the Theorem is non trivial. Remark 7.9.
It is known that the ambient Riemannian n -manifolds, n ≥ (with or without boundary)which contain a smooth m -dimensional submanifold, m ≥ , with null second fundamental form (i.e atotally geodesic submanifold) are quite rare. It could be interesting to show the same in the context ofvarifolds, that is to prove that the ambient compact Riemannian n -manifolds, n ≥ (with or withoutboundary) which contain a non zero (a priori non rectifiable) m -varifold, m ≥ , with null second fun-damental form relative to N (see Definition 2.5) are quite rare. This fact would imply the existence of alarger class of ambient Riemannian manifolds where the isoperimetric inequality Theorem 4.1 holds andthe main Theorem 1.3 is non trivial. References [1] W. K. Allard,
On the first variation of a varifold,
Annals of Math. Vol. 95, (1972), 417–491.[2] F. J. Almgren,
The theory of varifolds,
Mimeographed Notes, Princeton, (1965).[3] G. Anzellotti, R. Serapioni, I. Tamanini,
Curvatures, functionals, currents
Indiana Univ. Math. J.,Vol. 39, Num.3, (1990), 617–669.[4] L. Ambrosio, M. Gobbino, D. Pallara
Approximation Problems for Curvature Varifolds,
J. Geom.Anal. , Vol. 8, Num. 1, (1998), 1–19.[5] K. Brakke,
The Motion of a Surface by its Mean Curvature,
Mathematical Notes, Princeton Uni-versity Press (1978).[6] S. Delladio,
Special generalized Gauss graphs and their application to minimization of functionalsinvolving curvatures,
J. Reine Angew. Math., Vol. 486, (1997), 17–43.[7] J. P. Duggan, W ,p regularity for varifolds with mean curvature, Comm. Partial Diff. Eq., Vol. 11,Num. 9, (1986), 903–926.[8] H. Federer,
Geometric measure theory,
Springer, New York, (1969).[9] J. E. Hutchinson,
Second Fundamental Form for Varifolds and the Existence of Surfaces MinimizingCurvature,
Indiana Math. Journ., Vol. 35, Num. 1, (1986), 45–71.[10] J. E. Hutchinson, C ,α Multiple Function Regularity and Tangent Cone Behaviour for Varifolds withSecond Fundamental Form in L p , Proc. Symposia Pure Math., Vol. 44, (1986), 281–306.[11] J. E. Hutchinson,
Some regularity theory for curvature varifolds,
Miniconference geom. p.d.e. Proc.Center Math. An., Vol. 12, Australia Nat. Univ., (1987), 60–66.[12] E. Kuwert, R. Sch¨atzle,
Removability of isolated singularities of Willmore surfaces,
Annals of Math.Vol. 160, Num. 1, (2004), 315–357.[13] T. Lamm, J. Metzger, F. Schulze
Foliations of asymptotically flat manifolds by surfaces of Willmoretype , Math. Ann., Vol. 350, 1–78, (2011).[14] G. P. Leonardi, S. Masnou,
Locality of the mean curvature of rectifiable varifolds,
Adv. Calc. Var.,Vol. 2, Num. 1, (2009), 1742. 3315] C. Mantegazza,
Su Alcune Definizioni Deboli di Curvatura per Insiemi NonOrientati,
Degree Thesis at the Univ. of Pisa (1993), downloadable athttp://cvgmt.sns.it/cgi/get.cgi/papers/man93/thesis.pdf (Italian).[16] C. Mantegazza,
Curvature Varifolds with Boundary,
J. Diff. Geom., Vol. 43, (1996), 807–843.[17] U. Menne,
Second order rectifiability of integral varifolds of locally bounded first variation,
J. Geom.Anal., doi: 10.1007/s12220-011-9261-5, (2011).[18] U. Menne,
Some applications of the isoperimetric inequality for integral varifolds,
Adv. Calc. Var.,Vol. 2, (2009), 247–269.[19] W. H. Meeks, H. Rosenberg,
Stable minimal surfaces in M × R , J. Diff. Geom., Vol. 68, Num. 3,(2004), 515–534.[20] W. H. Meeks, H. Rosenberg,
The theory of minimal surfaces in M × R , Comm. Math. Helv., Vol.80, Num. 4, (2005), 811–858.[21] A. Mondino,
Some results about the existence of critical points for the Willmore functional,
Math.Zeit., Vol. 266, Num. 3, (2010), 583–622.[22] A. Mondino,
The conformal Willmore Functional: a perturbative approach, preprintarXiv:1010.4151v1 (2010), Jour. Geom. Anal. (24 September 2011), pp. 1-48, (Online First).[23] F. Morgan,
Geometric measure theory. A beginner’s guide. Fourth edition,
Elsevier/ AcademicPress, Amsterdam, (2009).[24] R. Moser,
A generalization of Rellich’s theorem and regularity of varifolds minimizing curvature,
Max Planck Inst. Math. Natur. Leipzig Preprint no. 72, (2001).[25] B. Nelli, H. Rosenberg,
Minimal surfaces in H × R , Bull. Braz. Math. Soc., Vol. 33, Num. 2, (2002),263–292.[26] R. Schoen, L. Simon,
Regularity of stable minimal hypersurfaces,
Comm. Pure Appl. Math., Vol.34, Num. 6, (1981), 741–797.[27] L. Simon,
Existence of surfaces minimizing the Willmore functional,
Comm. Anal. Geom., Vol. 1,Num. 2, (1993), 281–325.[28] L. Simon,
Lectures on geometric measure theory,
Proc. Center for Math. Analysis Australian Na-tional University, Vol.3, Canberra, Australia (1983).[29] J. T. Pitts,
Existence and regularity of minimal surfaces on Riemannian manifolds,
MathematicalNotes, Vol. 27, Princeton University Press, (1981).[30] T. Rivi`ere,
Analysis aspects of Willmore surfaces,
Invent. Math. Vol. 174, Num. 1, (2008), 1-45.[31] B. White,
Which ambient spaces admit isoperimetric inequalities for submanifolds?,
J. Diff. Geom.,Vol. 83, Num. 1, (2009), 213–228.[32] B. White,
The maximum principle for minimal varieties of arbitrary codimension, arXiv 0906.0189v2(2010).[33] T.J. Willmore,