Existence of stable H-surfaces in cones and their representation as radial graphs
aa r X i v : . [ m a t h . A P ] D ec EXISTENCE OF STABLE H-SURFACES IN CONES ANDTHEIR REPRESENTATION AS RADIAL GRAPHS
Paolo CALDIROLI, Alessandro IACOPETTI
Abstract.
In this paper we study the Plateau problem for disk-type surfaces contained inconic regions of R and with prescribed mean curvature H . Assuming a suitable growthcondition on H , we prove existence of a least energy H -surface X spanning an arbitraryJordan curve Γ taken in the cone. Then we address the problem of describing such surface X as radial graph when the Jordan curve Γ admits a radial representation. Assuming asuitable monotonicity condition on the mapping λ λH ( λp ) and some strong convexity-typecondition on the radial projection of the Jordan curve Γ, we show that the H -surface X canbe represented as a radial graph. Introduction
In the present paper we aim to investigate some aspects on the Plateau problem for disk-typesurfaces with prescribed mean curvature in the directions described as follows. Fixing a cone ofangular radius β C β := { p = ( x, y, z ) ∈ R | z > | p | cos β } , a Jordan curve Γ ⊂ C β \ { } , and a mapping H : C β → R , we are interested in finding conditionson H , possibly related to β , ensuring that stable surfaces in C β \ { } with mean curvature H ,spanning λ Γ do exist for every λ >
0. Moreover we address the problem of describing suchsurfaces as radial graphs when their boundaries admit a radial representation.In order to state our main results, let us state the analytical formulation of the problem. Let B = { ( u, v ) ∈ R | u + v < } be the unit open disk. In general, the Plateau problem for agiven Jordan curve Γ and a prescribed mean curvature function H consists in looking for maps X : B → R solving ∆ X = 2 H ( X ) X u ∧ X v in B (1.1) | X u | − | X v | = 0 = X u · X v in B (1.2) X | ∂B : ∂B → Γ is an (oriented) parametrization of Γ . (1.3)A map X ∈ C ( B, R ) ∩ C ( B, R ) satisfying (1.1)–(1.3) will be called H -surface spanning Γ(see [15]). It is known that that if X is an H -surface, then X has mean curvature H ( X ) apartfrom branch points, i.e., points ( u, v ) ∈ B where ∇ X ( u, v ) = 0.Our first result can be stated as follows. Mathematics Subject Classification.
Key words and phrases.
Prescribed mean curvature, Plateau’s problem, H-surfaces.
Acknowledgements.
Research partially supported by the project ERC Advanced Grant 2013 n. 339958 Com-plex Patterns for Strongly Interacting Dynamical Systems COMPAT, and by Gruppo Nazionale per l’AnalisiMatematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (IN-dAM). The first author is also supported by the PRIN-2012-74FYK7 Grant “Variational and perturbative aspectsof nonlinear differential problems”.
Theorem 1.1.
Let β ∈ (0 , π ) and let H : C β → R be a mapping of class C , satisfying | H ( p ) || p | cos β β ) ∀ p ∈ C β . (1.4) Then for every rectifiable Jordan curve Γ ⊂ C β \ { } there exists an H -surface X ∈ C ( B, R ) ∩ C ( B, R ) spanning Γ and contained in C β \ { } . Moreover we have that X ( B ) ⊂ C β . We point out that the assumption (1.4) fixes a bound on the radial behaviour of H withrespect to the angular diameter of the given Jordan curve Γ. Moreover, since (1.4) is asked tohold on a dilation-invariant domain and is independent of the curve Γ, the existence result statedby Theorem 1.1 remains true also taking λ Γ instead of Γ, for every λ >
0. Note that the case ofnonzero constant mean curvature is ruled-out.In fact we can provide more information on the H -surface given by Theorem 1.1. Moreprecisely, taking the variational character of the Plateau problem into account, such H -surfaceis characterized as a least energy surface, namely is a minimum point of the energy functionalassociated to system (1.1), in the class of admissible mappings satisfying (1.3). We refer toSections 2 and 3 for more details about this aspect.Our second result provides an answer to the issue of representing an H -surface as a radialgraph, when its contour is a radial graph. To this purpose, we need a monotonicity condition onthe mapping λ λH ( λp ) and some strong convexity-type condition on the radial projection ofthe Jordan curve Γ. In particular, we can show: Theorem 1.2.
Let β ∈ (0 , π ) and let H : C β → R be a mapping of class C ,α , satisfying (1.4)and H ( p ) + ∇ H ( p ) · p > ∀ p ∈ C β . (1.5) Let Γ be a regular Jordan curve of class C ,α contained in C β \ { } and let X be the least energy H -surface spanning Γ , given by Theorem 1.1. Assume that: (i) Γ is a radial graph, i.e. there exists a domain Ω ⊂ S and a map g : ∂ Ω → R + (with thesame regularity of Γ ) such that Γ = { g ( p ) p | p ∈ ∂ Ω } ; (ii) the domain Ω is β -convex (see Definition 4.2); (iii) the radial projection of X | ∂B induces a positive orientation on ∂ Ω (see Definition 5.3and Remark 5.5).Then the radial projection of X is a diffeomorphism between B and Ω and X ( B ) can be repre-sented as a radial graph. In particular X has no branch point. We notice that Theorem 1.2 is a corollary of a more general result (Theorem 6.1) about therepresentation of stable H -surfaces as radial graphs. The meaning of stable H -surface is explainedin Definition 2.7.The study developed in the present paper is the natural counterpart of analogous issues onthe Plateau problem for disk-type H -surfaces in a cylinder and their representation as cartesiangraphs with respect to the direction of the axis of the cylinder. On this side some results arealready known in in the literature: Rad´o proved in [10] that minimal surfaces, spanning a Jordancurve with one-one projection onto the boundary of a planar convex domain D ⊂ R , can berepresented as cartesian graphs of a function over D . Serrin in [13], Gulliver and Spruck in [8]proved the same result in the case of surfaces of constant mean curvature, but with differentassumptions. Sauvigny in [11] studied the case of stable H -surfaces with H not necessarilyconstant. In particular, he proved that, under a suitably strong convexity condition on theplanar domain D (which is the planar version of our β -convexity property), if H is monotonealong the axial direction of the cylinder, then stable H -surfaces can be represented as cartesiangraphs of a real valued function over D . -SURFACES IN CONES 3 Clearly our results cannot be recovered by those obtained for the cylinder. Indeed we dealwith a problem whose geometry exhibits a dilation invariance, in the sense that conditions (1.4)and (1.5) regard just the radial behaviour of the prescribed mean curvature function H .Furthermore, considering cones and radial projections rather that cylinders and cartesianprojections lead some non trivial, extra difficulties. The reason is that conical surfaces exhibit asingular point at their vertex and the radial projection is a nonlinear mapping. Let us displaythe main difficulties by highlighting the more delicate steps in our arguments.Concerning the existence result stated by Theorem 1.1, we follow the standard procedure ofminimizing the energy functional associated to the H -system (1.1) in the class of admissiblefunctions. Assumption (1.4) guarantees that the energy functional is bounded from below and,by known results, one gets existence of a minimizer X . Actually, in principle, the minimizercould touch the obstacle, in particular the vertex of the cone C β . To overcome this difficulty wesmooth the cone at the origin in a suitable way and we use a deep result by Gulliver and Spruck(see [9]), together with the growth condition (1.4), in order to obtain that the minimizer doesnot touch the boundary of the smoothed cone and then stays far from the vertex of C β . Thus,by well known regularity results (see for instance [4]), X is a classical solution of (1.1)–(1.3).Notice that our procedure needs more care than in the case of the analogous obstacle problemin a ball or in a cylinder (see Theorems 8 and 9 in Section 4.7 of [4]). We also observe that theminimizer X turns out to be stable in the sense of Definition 2.7, provided that Γ and H areregular enough (see also Proposition 2.3).Now let us spend a few words about our second result, concerning the characterization of astable H -surface X as radial graph, and let us shortly illustrate the strategy followed to showthat the radial projection P X = X/ | X | is a homeomorphism between B and Ω = P X ( B ).Under the assumptions on H and Γ as in the statement of Theorem 1.2, we prove that the radialcomponent of the Gauss Map N is always positive in B , namely N · X > B. (1.6)The maximum principle is the key tool to this aim. In fact, property (1.6) implies local invertibil-ity of P X far from branch points. The issue of global invertibility is not tackled with the samestrategy followed for the analogous problem of the projection along a fixed direction as in thepapers [8] and [11], because the expansion about branch points, based on the Hartman-Wintenertechnique, does not fit well with the radial projection. Instead, we follow an argument whichis mainly based on the degree theory, combined with a classical result about global invertibility(see [2]) and Jordan-Sch¨onflies’s Theorem (see [16]).Finally we notice that the (non-parametric) Plateau problem for H -surfaces characterizedas radial graphs was already discussed by Serrin in in [14]. Actually in that work a class ofpositively homogeneous prescribed mean curvature functions is considered and the existenceof ( n − H -surfaces in R n spanning a datum Γ is proved under the followingassumptions: Γ is the radial graph of a positive mapping f defined on the boundary of a givensmooth domain Ω contained in a hemisphere of S n , and H g ( y ) ≥ nn − H ( y ) f ( y ) ∀ y ∈ ∂ Ω , where H g denotes the geodesic mean curvature of ∂ Ω. For spherical caps, this condition turnsout to be less restrictive than (1.4). On the other hand, our results allow sign-changing andnon-homogeneous mean curvature functions, which cannot be considered in [14].Lastly, let us sketch an outline of the present paper: Sect. 2 contains a collection of knownfacts and technical results which will be used in the sequel. In Sect. 3 we prove the existenceresult stated by Theorem 1.1. In Sect. 4 we discuss the notion of β -convex domain in S . Finally -SURFACES IN CONES 4 Sections 5 and 6 contain the proof of (1.6) and of Theorem 1.2 (actually, a more general version),respectively. 2.
Notation and preliminary results
In this section we fix some notation and we collect some known facts which will be useful inthe rest of the paper.We denote by B the unit open disk of R and by B its closure. We will use indistinctly boththe real notation ( u, v ) or the complex notation z , w to denote a generic point of B or B . Inparticular, it will be always understood that z = e iθ ∈ ∂B stands for (cos θ, sin θ ). We denoteby S the unit sphere of R and by P : R \ { } → S the radial projection map, defined by P ( x ) := x | x | . We will use both the notation P ( Y ) or P Y to denote the composition P ◦ Y ,whenever Y is map with values in R \ { } .We begin with recalling some important facts about branch points and the normal N to anH-surface. Theorem 2.1 (see Theorem 1, Sect. 2.10, [4] and also Remark 3, Sect. 5.1, [3]) . Let X bean H-surface of class C ,α ( B, R ) or C ,α ( B, R ) respectively. Then, for each point w ∈ B or B , there is a vector A = ( A , A , A ) ∈ C with A + A + A = 0 , and a nonnegative integer n = n ( w ) such that X w ( w ) = A ( w − w ) n + o ( | w − w | n ) , as w → w , (2.1) where X w := ( X u − iX v ) . Remark 2.2.
The point w in the above statement is a branch point of X if and only if n ( w ) ≥ , and in this case n ( w ) is called the order of the branch point w ∈ B (or B respectively).Obviously w is regular point of X if and only if n ( w ) = 0 . Thanks to (2.1) we deduce thatbranch points of an H-surface are isolated. In particular, if X ∈ C ,α ( B, R ) then the set ofbranch points is finite. In order to get more analytic regularity on the solution X we have to ask more regularity onthe function H and on the Jordan curve Γ. More precisely, we recall that: Proposition 2.3 (see Chap. IX, Sect. 4, [12] and Sect. 2.3, [4]) . (i) If H ∈ C r,α ( R ) , for r ∈ N , α ∈ (0 , , then any solution X ∈ C ( B, R ) of (1.1) is ofclass C r +2 ,α ( B, R ) . (ii) If H ∈ C ,α ( R ) , for some α ∈ (0 , , and X is an H-surface such that X ( ∂B ) lieson a regular Jordan curve of class C ,α then X ∈ C ,α ( B, R ) . More in general, if H ∈ C r − ,α ( R ) and Γ ∈ C r,α , for some r ≥ , then X ∈ C r,α ( B, R ) . For X ∈ C ,α ( B, R ) we denote by B ′ the set of regular points. We recall that for an H-surface X ∈ C ,α ( B, R ) and w ∈ B ′ the normal N at w is given by N ( w ) = X u ( w ) ∧ X v ( w ) | X u ( w ) ∧ X v ( w ) | = X u ( w ) ∧ X v ( w ) | X u ( w ) | . (2.2)Thanks to the expansion (2.1), writing A = a − ib , with a, b ∈ R , from A = 0 and being A + A + A = 0 it follows that | a | = | b | 6 = 0, a · b = 0. Hence, if w is a branch point, then N ( w ) → a ∧ b | a | ∈ S , as w → w , w ∈ B ′ . Therefore we deduce that the normal N can be extended to a continuous function N ∈ C ( B, R ) with N ( B ) ⊂ S . Furthermore we have: -SURFACES IN CONES 5 Theorem 2.4 (see Theorem 1, Sect. 5.1, [3]) . Assume that H ∈ C ,α ( R ) and that X is anH-surface of class C ,α ( B, R ) . Then the normal N is of class C ,α ( B, R ) and satisfies thedifferential equation ∆ N + 2 pN = − E ∇ H ( X ) , (2.3) where E := | X u | , p := E [2 H ( X ) − K − ( ∇ H ( X ) · N )] , (2.4) is the so-called “density function” associated to X and K is the Gaussian curvature of X . More-over p ∈ C ,α ( B ) . As remarked in the introduction, it is well known that H-surfaces are obtained as stationarypoints of the energy functional F ( X ) = 12 Z B |∇ X | du dv + 2 Z B Q ( X ) · X u ∧ X v du dv, (2.5)where Q : R → R is a vector field such that div Q = H . Let us also introduce the functional G ( X ) := Z B | X u ∧ X v | du dv + 2 Z B Q ( X ) · X u ∧ X v du dv. Obviously we have F ( X ) ≤ G ( X ) and the equality F ( X ) = G ( X ) holds if and only X satisfiesthe conformality relations (1.2). Definition 2.5.
Let X an H-surface of class C ,α ( B, R ) and let ϕ ∈ C ∞ ( B ) be a test func-tion.We define the normal variation as the function Z : B × ( − ǫ , ǫ ) → R , ǫ > , definedby Z ( w, t ) := X ( w ) + tϕ ( w ) N ( w ) . We define the first variation and the second variation of G , in the normal direction Z , respectively,as δ G ( X, ϕN ) := ddt G ( Z ) (cid:12)(cid:12)(cid:12) t =0 and δ G ( X, ϕN ) := d dt G ( Z ) (cid:12)(cid:12)(cid:12) t =0 . The following result holds:
Theorem 2.6 (see Theorem 1, Sect. 5.3, [3]) . Let X ∈ C ,α ( B, R ) an H-surface and let ϕ ∈ C ∞ ( B ) a test function. Then: (i) δ G ( X, ϕN ) = 0 , (ii) δ G ( X, ϕN ) = Z B |∇ ϕ | − pϕ du dv , where p : B → R is given by (2.4) . We recall now the fundamental notion of stability for H-surfaces.
Definition 2.7.
We say that an H-surface X ∈ C ,α ( B, R ) is stable if it satisfies the followinginequality δ G ( X, ϕN ) ≥ , for all ϕ ∈ C ∞ ( B ) , which, in view of Theorem 2.6, can be rewritten as Z B |∇ ϕ | − pϕ du dv ≥ , for all ϕ ∈ C ∞ ( B ) . Remark 2.8.
We point out that global and local minimizers of G are stable. In particular if X satisfies the conformality relations and it is a minimizer of F , then it is stable. The following result is a well known version of the maximum principle. -SURFACES IN CONES 6
Proposition 2.9 (see Proposition 1, Sect. 5.3, [3]) . Assume that q ∈ C ,α ( B ) satisfies thestability inequality Z B |∇ ϕ | − qϕ du dv ≥ , for all ϕ ∈ C ∞ ( B ) , and let f ∈ C ( B ) ∩ C ( B ) be a solution of the boundary value problem ( ∆ f + 2 qf ≤ in Bf ( w ) > on ∂B. (2.6) Then f ( w ) > for all w ∈ B . Finally we state a classical result of global invertibility. We recall that a map F : X → Y between two topological spaces X , Y is said to be proper if F − ( K ) is compact in X , for anycompact subset K in Y . Theorem 2.10 (see Theorem 1.8, Sect. 3.1, [2]) . Let X , Y be two Banach spaces and let F : X → Y be a continuous surjective proper map. Suppose that F is locally invertible, X arcwise connected and Y simply connected. Then F is a homeomorphism. Existence of H -surfaces in cones In this section we prove Theorem 1.1. We divide the proof in several steps.
Step 1 : Extension of H to C β + δ , for δ > β ∈ (0 , π/
2) and set c β := cos β β ) . Let ¯ δ > β ± ¯ δ ∈ (0 , π/
2) and c β − δ < cotan( β + δ )2 for all 0 < δ < ¯ δ (this choice will be useful in the sequel of the proof). We point out that there always existsa ¯ δ = ¯ δ ( β ) > β ∈ (0 , π/
2) theinequality cos β β ) < cotan( β )2 is equivalent to sin β < β , which holds true. Thus, bycontinuity of the function δ cos( β − δ )2(1+cos( β − δ )) − cotan( β + δ )2 at 0 we get the desired assertion.Let 0 < δ < ¯ δ sufficiently small so that H can be extended to a function H ∈ C ( C β + δ ) with | H ( p ) || p | ≤ c β − δ (we observe that being γ c γ a strictly decreasing function it holds c β − δ > c β ).Clearly Γ is strictly contained in C β + δ . Step 2:
Construction of a suitable smooth surface of revolution which approximates ∂ C β + δ .The cone ∂ C β + δ is a non-smooth surface of revolution obtained by rotating the half-line σ ( t ) =(sin( β + δ ) t, , cos( β + δ ) t ), t ∈ R + ∪ { } , lying in the xz -plane, through the z -axis. We considerthe following approximating surface of revolution: let ǫ > t ǫ := √ β + δ ) ǫ , let S β + δ,ǫ be the surface obtained by rotating the curve σ ǫ ( t ) :=( α ( t ) , , α ( t )) through the z -axis, parametrized by φ ( t, θ ) = ( α ( t ) cos θ, α ( t ) sin θ, α ( t )),where α ( t ) := sin( β + δ ) t, t ≥ ,α ( t ) := ( a ǫ t + b ǫ t + c ǫ , if t ∈ [0 , t ǫ ] , cos( β + δ ) t, if t ∈ ] t ǫ , + ∞ [ . (3.1)with a ǫ , b ǫ , c ǫ chosen in a suitable way in order that:(i) S β + δ,ǫ is of class C , -SURFACES IN CONES 7 (ii) 0 S β + δ,ǫ ,(iii) the component S β + δ,ǫ of R \ S β + δ,ǫ which does not contain the origin is convex,(iv) the mean curvature (with respect to the inward normal) H S β + δ,ǫ of S β + δ,ǫ satisfies | H ( p ) | < H S β + δ,ǫ ( p ) , for any p ∈ S β + δ,ǫ . (3.2)A good choice of the coefficients a ǫ , b ǫ , c ǫ is a ǫ := −√ (cid:18) (cid:19) cos ( β + δ ) ǫ , b ǫ := 2 √ (cid:18) (cid:19) cos ( β + δ ) ǫ , c ǫ := ǫ √ . Inequality (3.2) is checked at Step 8.
Step 3:
Choice of a vector field Q : C β + δ → R such that div Q = H .Let us set Q ( p ) := (cid:18)Z H ( tp ) t dt (cid:19) p, p ∈ C β + δ . It is clear that Q ∈ C ( C β + δ , R ) and by elementary computations we see that div Q = H in C β + δ . Moreover we observe that, since | H ( p ) || p | ≤ c β − δ for all p ∈ C β + δ , we have that k Q k ∞ , C β + δ ≤ c β − δ < . (3.3) Step 4:
Construction of a weak solution of (1.1) which satisfies (1.2) a.e. in B .Let ǫ > ⊂ S β + δ,ǫ and (3.2) holds. We consider the variationalproblem P (Γ , S β,ǫ ) given by min X ∈C (Γ , S β + δ,ǫ ) F ( X ) , where F is the functional defined in (2.5) and C (Γ , S β + δ,ǫ ) is the class of the admissible functions,i.e., the set of the functions in H , ( B, R ) ∩ C ( ∂B, R ) which map ∂B weakly monotonic ontoΓ, satisfy a three point condition and have an image almost everywhere in S β + δ,ǫ (see also [3]).Since Q verifies (3.3) and S β + δ,ǫ ⊂ C β + δ we get that F is coercive. In fact, considering theassociated Lagrangian e ( p, q ) = 12 ( q + q ) + 2 Q ( p ) · q ∧ q , where p = ( x, y, z ) ∈ S β + δ,ǫ , q = ( q , q ) ∈ R × R , by elementary computations and using(3.3), we get that, for any p ∈ S β + δ,ǫ , q ∈ R × R (cid:18) − c β − δ (cid:19) ( | p | + | p | ) ≤ e ( p, q ) ≤ (cid:18)
12 + c β − δ (cid:19) ( | p | + | p | ) . In order to minimize the energy functional we have to prove that the class of admissiblefunctions is not empty, i.e., C (Γ , S β + δ,ǫ ) = ∅ . To this end we recall that since Γ is rectifiable itis well known that the set C (Γ , R ) is not empty (see [3] pag. 255) and there exists a minimalsurface Y ∈ C (Γ , R ) spanning Γ. Since Y ∈ C ( B, R ) ∩ C ( B, R ) is harmonic, by the Convexhull theorem (see Theorem 1, Section 4.1 of [4]) we have that Y ( B ) is contained in the convex hullof Γ. In particular, being S β + δ,ǫ convex we get that Y ( B ) ⊂ S β + δ,ǫ . Hence Y ∈ C (Γ , S β + δ,ǫ ).By Theorem 3 in Section 4.7 of [4] we have that the variational problem P (Γ , S β + δ,ǫ ) has aweak solution X ∈ C (Γ , S β + δ,ǫ ) and it satisfies the conformality relations | X u | = | X v | , X u · X v = 0 a.e. in B. Step 6:
The weak solution X found at Step 5 is a classical solution of (1.1) and maps homeo-morphically ∂B onto Γ.Since S β + δ,ǫ is a closed and convex set, such that ˚ S β + δ,ǫ = S β + δ,ǫ , we have that S β + δ,ǫ is a -SURFACES IN CONES 8 quasi-regular set (see Remark (i), pag 381 in [4]). Thanks to a well known regularity result (seeTheorem 4, pag 381 in [4]) since S β + δ,ǫ is quasi-regular it follows that X is continuous up tothe boundary. In order to get more regularity and prove that X is a classical solution of (1.1),we show that X does not touch the boundary ∂ S β + δ,ǫ = S β + δ,ǫ . To prove this, we will argue bycontradiction and use an important result of Gulliver and Spruck, which is a sort of geometricmaximum principle.Assume by contradiction that X touches S β + δ,ǫ . The idea is to show that in this case we canconstruct Y ∈ C (Γ , S β + δ,ǫ ) such that F ( Y ) < F ( X ), and hence, being X of least energy we geta contradiction. To this end we define a “truncation” map T : S β + δ,ǫ → R .In order to define T we need some preliminary definitions: for p ∈ S β + δ,ǫ we define r ( p ) := dist ( p, S β + δ,ǫ ), we observe that there exists a neighborhood V of S β + δ,ǫ such that for p ∈ V thereis a unique point π ( p ) ∈ S β + δ,ǫ with | p − π ( p ) | = r ( p ). We observe that, in the definition of π it isfundamental that S β + δ,ǫ is smooth: in fact, in the case of a cone, for any neighborhood V of thecone, we have that any point p ∈ V lying on the axis of the cone we have that p is equidistantfrom S β + δ,ǫ , so π ( p ) cannot be defined as in the previous way.We also observe that π : V → S β + δ,ǫ is a C map. Finally, for R > T : S β + δ,ǫ → R by setting T ( p ) := ( π ( p ) + RN ( π ( p )) if p ∈ V and r ( p ) ≤ R,p otherwise , where N ( q ) is the inward normal at q ∈ S β + δ,ǫ . In general T may be not continuous, but thanksto Theorem 3.1 of [9], since (3.2) holds, there exists R > < R ≤ R andinf z ∈ B r ( X ( w )) < R we have T ◦ X ∈ C ( B ) ∩ H , ( B, R ) and F ( T ◦ X ) < F ( X ). Since we areassuming that X touches S β + δ,ǫ we have that inf z ∈ B r ( X ( w )) < R , for any 0 < R < R . Hence F ( T ◦ X ) < F ( X ). It remains to prove that T ◦ X ∈ C (Γ , S β + δ,ǫ ). From the proof of Theorem3.1 in [9] we know that T ◦ X ∈ C ( B ) ∩ H , ( B, R ), moreover since Γ is strictly in the interiorof S β + δ,ǫ , for R sufficiently small, by definition of T we have that T ( p ) = p , for any p ∈ Γ.Hence, since X is a weakly monotonic map of ∂B onto Γ, and satisfies a three point condition,the same holds for T ◦ X , and thus T ◦ X ∈ C (Γ , S β + δ,ǫ ) and we get the contradiction.Therefore we have that X ( B ) ∩ S β + δ,ǫ = ∅ , so from Theorem 7 in Section 4.7 of [4] wehave that X is a classical solution of (1.1) and X ( B ) ⊂ S β + δ,ǫ . Moreover we observe that byconstruction, for all sufficiently small ǫ > S β + δ,ǫ ⊂ C β + δ .We observe that X : ∂B → Γ is a homeomorphism. This follows in a standard manner (seefor instance the proof of Theorem 8 and the Remark at page 402 in [4]).
Step 7: X ( B ) ⊂ C β .We begin with proving that X ( B ) ⊂ C β \{ } . Let us set φ ( u, v ) := X ( u, v ) · e −| X ( u, v ) | cos β ,where e = (0 , , φ ≥ B . To this end we first show that − ∆ φ ≥ B , i.e., φ is super-harmonic in B .By elementary computations we have that φ u = X u · e − X · X u | X | cos β , and φ uu = X uu · e − X u + X · X uu | X | cos β + ( X · X u ) | X | cos β. Hence we get that − ∆ φ = − ∆ X · e + 2 E + X · ∆ X | X | cos β − ( X · X u ) + ( X · X u ) | X | cos β, -SURFACES IN CONES 9 where E = | X u | = | X v | . Since 0 X ( B ) we have that φ ∈ C ( B ) ∩ C ( B ). Now, recallingthat X is an H-surface, we deduce that in the subset B ′ ⊂ B of regular points it holds − ∆ φ = − ∆ X · e + 2 E + X · ∆ X | X | cos β − ( X · X u ) + ( X · X v ) | X | cos β = − H ( X )( X u ∧ X v · e ) + 2 E | X | cos β + 2 H ( X )( P ( X ) · X u ∧ X v ) cos β − ( P ( X ) · P ( X u )) + ( P ( X ) · P ( X v )) | X | E cos β ≥ − | H ( X ) | E + 2 E | X | cos β − | H ( X ) | E cos β − E | X | cos β = cos β − β ) | H ( X ) || X || X | E ≥ . (3.4)We point out that the last inequality holds because H satisfies assumption (1.4), and the pre-vious one is a consequence of ( P ( X ) · P ( X u )) + ( P ( X ) · P ( X u )) ≤
1, which comes from theorthogonality of the versors P ( X u ), P ( X v ).On the other hand, if ( u , v ) ∈ B is a branch point then, from the first line of (3.4) we getthat − ∆ φ ( u , v ) = 0. Hence we have proved that − ∆ φ ≥ B and we are done.Now, since X maps ∂B onto Γ and Γ ⊂ C β \ { } we have that φ ≥ ∂B . Therefore, by themaximum principle we get that φ ≥ B , from which we get that X ( B ) ⊂ S β,ǫ ⊂ C β \ { } .Now, from Enclosure Theorem I (see Section 4.2 in [4]), in view of (3.2) (which holds for δ = 0), we get that X ( B ) ⊂ S β,ǫ , from which we deduce that X maps B into C β , and we aredone. Step 8:
Proof of (3.2).Using the parametrization φ ( t, θ ) = ( α ( t ) cos θ, α ( t ) sin θ, α ( t )) we have that the mean curva-ture (with respect to the inward normal) of S β + δ,ǫ is given by H S β + δ,ǫ = α ( α ′ α ′′ − α ′ α ′′ ) + α ′ (( α ′ ) + ( α ′ ) )2 α (( α ′ ) + ( α ′ ) ) / (see for instance [1]). For t > t ǫ we have that the rotation of σ ǫ ( t ) = ( α ( t ) , , α ( t )) describes aportion of the cone ∂ C β + δ and it is elementary to see that H S β + δ,ǫ ( t ) = cotan( β + δ )2 t . On the other hand, if p = φ ( t, θ ) ∈ S β + δ,ǫ , for t > t ǫ , θ ∈ [0 , π [ we have | H ( p ) | ≤ c β − δ | φ ( t, θ ) | = c β − δ p α ( t ) + α ( t ) = c β − δ t . Hence | H ( φ ( t, θ )) | < H S β + δ,ǫ ( t ) , for any t > t ǫ , θ ∈ [0 , π [ (3.5)if and only if c β − δ < cotan( β + δ )2 , which holds true as displayed in Step 1.For the remaining interval [0 , t ǫ ] we have the following:lim ǫ → min t ∈ [0 ,t ǫ ] H S β + δ,ǫ ( t ) = + ∞ . (3.6) -SURFACES IN CONES 10 Before proving (3.6) we observe that it implies that there exists a small ¯ ǫ > k H k ∞ , C β + δ ∩{ z ≤ } < min [0 ,t ǫ ] H S β + δ,ǫ ( t ) for all 0 < ǫ < ¯ ǫ . Hence for all sufficiently small ǫ > H ( φ ( t, θ )) < H S β + δ,ǫ ( t ) , for any t ∈ [0 , t ǫ ] , θ ∈ [0 , π [ . (3.7)At the end, thanks to (3.5), (3.7), we get (3.2).Now we prove (3.6). First, for ǫ > t ∈ [0 , t ǫ ], we have that H S β + δ,ǫ ( t ) ≥ − √ (cid:0) (cid:1) ( β + δ ) ǫ t + 4 √ (cid:0) (cid:1) ( β + δ ) ǫ β + δ ) (cid:16) (sin( β + δ )) + ( − √ (cid:0) (cid:1) ( β + δ ) ǫ t + 4 √ (cid:0) (cid:1) ( β + δ ) ǫ t ) (cid:17) / . (3.8)In fact observe that α ′′ ≡ α ≥ α ′ > α ′′ > , t ǫ ], hence H S β + δ,ǫ ( t ) = α ( α ′ α ′′ − α ′ α ′′ ) + α ′ (( α ′ ) + ( α ′ ) )2 α (( α ′ ) + ( α ′ ) ) / ≥ α ( α ′ α ′′ ) + α ′ (( α ′ ) + ( α ′ ) )2 α (( α ′ ) + ( α ′ ) ) / ≥ α ′ α (( α ′ ) + ( α ′ ) ) / that is (3.8). Now, setting s := tǫ , g ǫ ( s ) := − √ (cid:0) (cid:1) cos ( β + δ ) s ǫ + 4 √ (cid:0) (cid:1) cos ( β + δ ) ǫ , h ( s ) := − √ (cid:0) (cid:1) cos ( β + δ ) s + 4 √ (cid:0) (cid:1) cos ( β + δ ) s by the previous estimate we deducethat min t ∈ [0 ,t ǫ ] H S β + δ,ǫ ( t ) ≥ min s ∈ h , √ β + δ ) i g ǫ ( s )2 sin( β + δ ) ((sin( β + δ )) + ( h ( s )) ) / . Since s ∈ h , √ β + δ ) i , g ǫ ( s ) = 4 √ (cid:0) (cid:1) cos ( β + δ ) (cid:16) − (cid:0) (cid:1) cos ( β + δ ) s ǫ + ǫ (cid:17) , it is ele-mentary to see that g ǫ ( s ) ≥ √ (cid:18) (cid:19) cos ( β + δ ) 23 ǫ . Moreover, since s
12 sin( β + δ )((sin( β + δ )) +( h ( s )) ) / does not depend on ǫ , there exists a positiveconstant C depending only on β + δ such that, for any s ∈ h , √ β + δ ) i , we have12 sin( β + δ ) ((sin( β + δ )) + ( h ( s )) ) / > C . Finally, putting together these estimates, we havemin t ∈ [0 ,t ǫ ] H S β + δ,ǫ ( t ) ≥ C √ (cid:18) (cid:19) cos ( β + δ ) 23 ǫ → + ∞ , as ǫ → . Hence (3.6) is proved.The proof is now complete.4. On β -convex domains and related results In this section we introduce the definition of β -convexity and prove some geometric resultsabout β -convex subsets of S as well as geometric results about H -surfaces having support ina cone, whose boundary datum radially projects onto the boundary of a smooth β -convex subset.Let Ω be a open subset of the unit sphere S such that ∂ Ω is a Jordan curve. We denote by C Ω the conic region in R spanned by Ω. -SURFACES IN CONES 11 Definition 4.1.
We say that Ω is convex if C Ω is a convex subset of R . In order to get our results we need of a stronger convexity notion. For ˆ p ∈ S and β ∈ (0 , π )we set C ˆ p ,β := { x ∈ R ; x · ˆ p − | x | cos β > } . We introduce the following definition:
Definition 4.2.
Let β ∈ (0 , π ) . We say that Ω verifies a β -cone condition at a given p ∈ ∂ Ω ifthere exists ˆ p ∈ S such that p ∈ ∂ C ˆ p ,β and Ω ⊂ C ˆ p ,β . We say that Ω is β -convex if, for any p ∈ ∂ Ω , Ω verifies a β -cone condition at p . We observe that, by definition, if Ω is β -convex, then, it is strictly contained in a hemisphere.At first sight, one could think that for any p ∈ ∂ Ω there could be many ˆ p ∈ S satisfying the β -cone condition at p , but this is not the case: Proposition 4.3.
Assume that Ω verifies a β -cone condition at p ∈ ∂ Ω and that ∂ Ω is a regularJordan curve of class C . Then there exists only one ˆ p ∈ S such that p ∈ ∂ C ˆ p ,β and Ω ⊂ C ˆ p ,β .Moreover the mapping p ˆ p is continuous from ∂ Ω into S .Proof. Let σ : ( − δ, δ ) → ∂ Ω be a C -parametrization of a portion of ∂ Ω, centered at p . Since ∂ Ωis a regular curve we can assume that σ ′ ( t ) = 0 in ( − δ, δ ). Since Ω verifies a β -cone conditionat p , then, all possible ˆ p = ˆ p ( p, β ) lie in ∂ C p,β ∩ S . Now observe that for any admissible ˆ p ,since σ (0) · ˆ p = cos β , σ ( t ) · ˆ p ≥ cos β in ( − δ, δ ), then, the function h ( t ) := σ ( t ) · ˆ p must havenull derivative at 0. Hence σ ′ (0) · ˆ p = 0, which means that all possible ˆ p ( p, β ) must lie in theplane { σ ′ (0) } ⊥ . We also observe that since | σ | ≡
1, then, by deriving this relation, we get that p ∈ { σ ′ (0) } ⊥ .Thus all possible ˆ p are given by the intersection ∂ C p,β ∩ { σ ′ (0) } ⊥ ∩ S which consists of twovectors ˆ p , , ˆ p , . By construction we observe that they generate two cones ∂ C ˆ p , ,β , ∂ C ˆ p , ,β such that ∂ C ˆ p , ,β ∩ ∂ C ˆ p , ,β = { λp, λ ∈ R + } . Hence, since Ω must be entirely contained in oneof the regions C p , ,β , C p , ,β , we have that only one of the two vectors ˆ p , , ˆ p , is admissible.The first part of proof is then complete.We prove now the continuity of the map p ˆ p , from ∂ Ω into S . If σ : ( − δ, δ ) → ∂ Ω is alocal parametrization centered at p ∈ ∂ Ω, then, as seen in the first part of the proof we haveˆ p ( σ ( t )) = ∂ C σ ( t ) ,β ∩ { σ ′ ( t ) } ⊥ ∩ S ∩ Ω. Hence, it is clear that ˆ p ( σ ( t )) depends continuously on t and we are done.The proof is then complete. (cid:3) Next proposition states that β -convexity is actually a convexity property. Proposition 4.4. If Ω is β -convex then Ω is convex.Proof. Assume by contradiction that Ω is not convex. Then, there exist two distinct points p , p ∈ C Ω such that the segment σ ( t ) joining p and p is not entirely contained in C Ω . Let usset ˆ p := P ( p ), ˆ p := P ( p ), ˆ σ := P ◦ σ . Then, there exists t ∈ (0 ,
1) such that ˆ σ ( t ) ∈ ∂ Ω.Since Ω is β -convex, choosing p := ˆ σ ( t ) in the definition, we get that there exists ˆ p such thatΩ is contained in the region C ˆ p ,β , and p ∈ ∂ Ω ∩ ∂ C ˆ p ,β . We observe that since ˆ p , ˆ p ∈ Ω thenˆ p , ˆ p ∈ C ˆ p ,β (they cannot lie on its boundary ∂ C ˆ p ,β , otherwise they would belong to ∂ Ω).Hence we have that p , p ∈ C ˆ p ,β but σ ( t ) C ˆ p ,β which contradicts the convexity of C ˆ p ,β . (cid:3) Now let us examine the relationship between the notion of β -convexity and some geometricalproperties of H -surfaces. We begin with the following preliminary result: -SURFACES IN CONES 12 Proposition 4.5.
Let β ∈ (0 , π ) , let Ω ⊂ S be a β -convex domain and let Γ be a smooth regularJordan curve such that P (Γ) ⊂ ∂ Ω . Assume that H satisfies (1.4) , and let X ∈ C ( B, R ) ∩ C ( B, R ) be an H-surface, with X ( B ) ⊂ C β \ { } . Then, for any p ∈ ∂ Ω , the associated function φ p ( u, v ) := X ( u, v ) · ˆ p − | X ( u, v ) | cos β is strictly positive in B , where ˆ p = ˆ p ( p, β ) ∈ S is givenby the definition of β -convexity.Proof. Let us fix p ∈ ∂ Ω. Since Ω is β -convex there exists ˆ p ∈ S such that p ∈ ∂ C ˆ p ,β and Ω iscontained in C ˆ p ,β . Hence, setting φ p ( u, v ) := X ( u, v ) · ˆ p − | X ( u, v ) | cos β , we have that φ p ≥ ∂B . By replacing e with ˆ p in the proof of Step 7 we have that φ p is super-harmonic in B ,and by the maximum principle we get that φ p ≥ B . From the strong maximum principle itfollows that φ p > B or φ p ≡ B . To complete the proof we have to show that the latterpossibility cannot occur.Assume by contradiction that φ p ≡ B , then, by definition and since X is smooth wehave that X ( B ) ⊂ ∂ C ˆ p ,β \ { } . Without loss of generality we can assume that ˆ p = e sothat X ( B ) is entirely contained in the surface ∂ C β \ { } which is the surface of revolutiongenerated by the rotation, with respect of the z -axis, of the curve σ , lying in the xz -plane,given by σ ( t ) := ( α ( t ) , , α ( t )), where α ( t ) = sin( β ) t , α ( t ) = cos( β ) t , t >
0. As seen inthe proof of Theorem 1.1, using the parametrization φ ( t, θ ) = ( α ( t ) cos θ, α ( t ) sin θ, α ( t )), wehave that the mean curvature of ∂ C β \ { } (with respect to the inward normal) is given by H ∂ C β \{ } ( t ) = t cotan( β )2 , t >
0, moreover | H ( φ ( t, θ )) | < H ∂ C β \{ } ( t ) for all t > θ ∈ [0 , π ]. Infact, since H satisfies (1.4), then for all p = φ ( t, θ ) ∈ ∂ C β \ { } we have | H ( φ ( t, θ )) | ≤ c β | φ ( t, θ ) | = c β t < t cotan( β ) = H ∂ C β \{ } ( t ) , (4.1)because c β = cos β β ) < cotan β . Thanks to (4.1), Theorem 2 and Corollary 3 in Section 4.4of [4] (or by Enclosure Theorem I in Section 4.2 of [4]) it follows that X ( B ) ∩ ( ∂ C β \ { } ) = ∅ which gives a contradiction. The proof is then concluded. (cid:3) Corollary 4.6.
Under the same assumptions of the previous proposition we have that, for any ( u, v ) ∈ ∂B , the normal derivative, with respect to the exterior normal ν of the function φ p ,corresponding to p = P X ( u, v ) ∈ ∂ Ω , is strictly negative at ( u, v ) , i.e. ∂∂ν φ p ( u, v ) < . In particular, if X ∈ C ( B, R ) then X has no boundary branch points.Proof. Let us fix ( u, v ) ∈ ∂B and let p = P X ( u, v ) ∈ ∂ Ω. Consider the associated function φ p .As seen in the proof of Proposition 4.5 we have that φ p is super-harmonic in B and φ p > φ p ( u, v ) = 0, by Hopf’s Lemma, we get that ∂∂ν φ p ( u, v ) <
0, where ν = ( ν , ν )denotes the exterior normal at ( u, v ) ∈ ∂B . The first part is then proved.For the second part we observe that since ∂φ p ∂u = X u · ˆ p − X · X u | X | cos β , we have ∂∂ν φ p ( u, v ) = ( X u · ˆ p ) ν + ( X v · ˆ p ) ν − ( X · X u ) ν + ( X · X v ) ν | X | cos β < . Since ( u, v ) is arbitrary we get that X cannot have branch points on ∂B . (cid:3) Another important and immediate consequence of Proposition 4.5 is the following:
Proposition 4.7.
Under the same assumptions of Proposition 4.5 we have that
P X ( B ) ⊂ Ω . In particular X has support in the cone spanned by Ω , i.e., X ( B ) ⊂ C Ω \ { } . -SURFACES IN CONES 13 Proof.
Assume by contradiction that there exists some ( u , v ) ∈ B such that P X ( u , v ) ∈ S \ Ω,then, necessarily, there exists ( u , v ) ∈ B such that X ( u , v ) ∈ ∂ C Ω \ { } .In fact, on the contrary, we would have that X ( B ) ∩ ( ∂ C Ω \ { } ) = ∅ , and hence we wouldhave X ( B ) = ( X ( B ) ∩ C Ω ) ∪ [ X ( B ) ∩ ( R \ C Ω )] . Since we are assuming that
P X ( u , v ) ∈ S \ Ω, we have that both the open sets in the right-handside are nonempty, hence, since they are disjoint and X ( B ) is connected we get a contradiction.Hence there exists ( u , v ) ∈ B such that X ( u , v ) ∈ ∂ C Ω \{ } , and taking p = P X ( u , v ) ∈ ∂ Ω, by the definition of β -convexity and applying Proposition 4.5 to the function φ p , we get acontradiction since φ p ( u , v ) = 0. (cid:3) Stable H-surfaces with one-one radial projection onto a β -convex subset In this section we analyze the geometrical properties of stable H -surfaces whose boundary isa Jordan curve Γ that projects bijectively onto the boundary of a smooth β -convex domain Ωof the unit sphere S . It will be understood, if not specified, that Γ is a Jordan curve of class C ,α and H ∈ C ,α , for some α ∈ (0 , C ,α ( B, R ) (see also Proposition 2.3).We begin with a preliminary proposition: Proposition 5.1.
Let X be a stable H-surface of class C ,α ( B, R ) , with H satisfying (1.5) .Assume that N · X > on ∂B , then N · X > in B .Proof. Let us set f := N · X . By elementary computations we have f u = N u · X + N · X u = N u · X , and thus f uu = N uu · X + N u · X u . Deriving the relation N · X u ≡ N u · X u = − N · X uu . Hence f uu = N uu · X − N · X uu and thus ∆ f = ∆ N · X − N · ∆ X . Now,thanks to Theorem 2.4, in the subset B ′ ⊂ B of regular points, we get that∆ f + 2 pf = − E ∇ H ( X ) · X − H ( X )[ N · ( X u ∧ X v )]= − E ( ∇ H ( X ) · X + H ( X )) . Since we are assuming (1.5) we have − E ( ∇ H ( X ) · X + H ( X )) ≤ B ′ .On the other hand in the subset of branch points of X we have ∆ f + 2 pf = − E ∇ H ( X ) · X − H ( X )[ N · ( X u ∧ X v )] = 0. Now applying Proposition 2.9 (we recall that p ∈ C ,α ( B )) we getthat f > B and we are done. (cid:3) It remains to study the sign of N · X on the boundary ∂B . The next proposition ensures that N · X never vanishes on ∂B . Proposition 5.2.
Let Ω be a β -convex domain of class C ,α and let Γ be a Jordan curve of class C ,α which radially projects onto ∂ Ω . Assume that X is an H-surface of class C ,α ( B, R ) with H satisfying (1.4) . Then the function N · X never vanishes on ∂B , hence N · X has a constantsign on ∂B .Proof. Let us set f := N · X . Assume by contradiction that there exists z ∈ ∂B such that f ( z ) = 0. In particular, since X has no boundary branch points (see Corollary 4.6) then wehave X ( z ) · X u ( z ) ∧ X v ( z ) = 0. This means that X ( z ) ∈ Span { X u ( z ) , X v ( z ) } := Π. Henceit follows that ( P X ) u ( z ) = X u ( z ) | X ( z ) | − X ( z ) · X u ( z ) | X ( z ) | X ( z ) ∈ Π , and the same happens for ( P X ) v ( z ). Moreover, by deriving | P X | ≡
1, we get that
P X · ( P X ) u ≡ P X · ( P X ) v ≡ -SURFACES IN CONES 14 Let us set v := P X ( z ), v := ( P X ) u ( z ), v := ( P X ) v ( z ) and let us observe that v , v , v ∈ Π and v · v = v · v = 0. In particular, since v = 0 we deduce that v ∧ v = 0 . (5.1)Let φ = X · ˆ p − | X | cos β be the associated function to v ∈ ∂ Ω. In particular we have φ ( z ) =0, φ ≥ B and ∂φ∂ν ( z ) < ψ : R → R ,defined by ψ ( θ ) := P X (cos θ, sin θ ) · ˆ p − cos β and let θ ∈ [0 , π [ such that z = (cos θ , sin θ ).Since ψ ≥ ψ ( θ ) = 0 and ψ ∈ C ( R ) we have that ψ ′ ( θ ) = 0. This means that v · ˆ p ( − ν ) + v · ˆ p ( ν ) = 0 , (5.2)where ν = cos θ , ν = sin θ . Moreover, since φ ( z ) = 0 we observe that ∂φ∂ν ( z ) < v · ˆ p ( ν ) + v · ˆ p ( ν ) < . (5.3)We show that (5.1), (5.2) and (5.3) lead to a contradiction.If v = 0 and v = 0 then, setting a := v · ˆ p , b := v · ˆ p we rewrite (5.2), (5.3) as ( − ν a + ν b = 0 ν a + ν b = − k, for some k >
0. Then, by elementary computations it follows that ( a, b ) = − k ( ν , ν ) . Hence wehave that ( v · ˆ p = − kν ,v · ˆ p = − kν . (5.4)On the other hand v ∧ v = 0 implies that v = λv , for some λ = 0, and hence from (5.4) wehave − kν = λv · ˆ p = λ ( − kν ) . (5.5)Remembering that (1.1) is invariant under conformal transformations of the unit disk into itself,up to a rotation of angle 2 π − θ , we can assume that z = (1 , ν = 1, ν = 0and so, since k = 0, we contradicts (5.5).It remains to examine the case in which at least one between v , v is zero. Assume bycontradiction that v = 0, then, thanks to (5.2), (5.3) we get that ( v · ˆ p ( ν ) = 0 v · ˆ p ( ν ) < , (5.6)Up to a rotation we can assume that ν = 0, ν = 0, and hence (5.6) gives a contradiction. Thesame argument shows that v = 0 cannot happen. The proof is complete. (cid:3) It remains to prove that N · X > ∂B . To this end we we introduce the following definition: Definition 5.3.
Let Ω ⊂ S be a β -convex domain, such that ∂ Ω is a regular Jordan curve ofclass C , i.e., there exists a parametrization γ : ∂B → ∂ Ω of class C which is a homeomorphismand satisfies γ ′ ( z ) = 0 for all z = e iθ ∈ ∂B , where γ ′ ( z ) = ddθ γ ( e iθ ) . We say that ∂ Ω is positivelyoriented by γ if we have ( γ ′ ( z ) ∧ γ ( z )) · ˆ p ( z ) < , for all z = e iθ , θ ∈ [0 , π [ , where ˆ p ( z ) is theversor associated to γ ( z ) , given by the definition of β -convexity. Remark 5.4.
We point out that the sign of ( γ ′ ( z ) ∧ γ ( z )) · ˆ p ( z ) is well defined since, as provedin Proposition 4.3, there is only one ˆ p ( z ) ∈ S satisfying the β -convexity condition at γ ( z ) ∈ ∂ Ω .Moreover, for any z ∈ ∂B , it cannot happen that ( γ ′ ( z ) ∧ γ ( z )) · ˆ p ( z ) = 0 . In fact, if we considerthe scalar function θ h ( θ ) := γ ( e iθ ) · ˆ p , since h has a minimum at θ corresponding to z , weget that γ ′ ( z ) · ˆ p = 0 and hence if by contradiction ˆ p ( z ) ∈ Span { γ ′ ( z ) , γ ( z ) } , then ˆ p would beproportional to γ ( z ) which is not possible. Hence we must have Det [ γ ′ ( z ) , γ ( z ) , ˆ p ( z )] = 0 on ∂B . -SURFACES IN CONES 15 Furthermore, thanks to the second part of Proposition 4.3, we deduce that
Det [ γ ′ ( z ) , γ ( z ) , ˆ p ( z )] is continuous on ∂B . Hence Definition 5.3 well defines an orientation on ∂ Ω . Now we have all the instruments to state our assumption, which will be crucial for gettingour next results.Given an H -surface X of class C ,α ( B, R ) spanning a regular Jordan curve Γ of class C ,α we introduce the following: Assumption (I) :(i) Γ is a radial graph, i.e. there exists a domain Ω ⊂ S and a map g : ∂ Ω → R + (with thesame regularity of Γ) such that Γ = { g ( p ) p | p ∈ ∂ Ω } ;(ii) the domain Ω is β -convex;(iii) the radial projection of X | ∂B induces a positive orientation on ∂ Ω. Remark 5.5.
We observe that, in our context, assumption (iii) makes sense. In fact, by defi-nition of H -surface we have that X | ∂B : ∂B → Γ is an homeomorphism and by Corollary 4.6 weknow that X has no boundary branch points. Proposition 5.6.
Let Γ be a regular Jordan curve of class C ,α contained in C β \ { } and let X ∈ C ,α ( B, R ) be an H -surface spanning Γ . Suppose that Assumption (I) is satisfied. Then N · X > on ∂B .Proof. Let z = ( u , z ) ∈ ∂B the point in which | X | achieves its maximum and set M :=sup p ∈ Γ | p | . Let ˆ p ∈ S be the versor associated to P X ( z ) by the definition of β -convexity. Upto a rotation of angle θ ∈ [0 , π [ we can assume that z = (1 , ∂ Ω. Thanks to Corollary 4.6, since ν = (1 , X u ( z ) · ˆ p < X ( z ) · X u ( z ) | X ( z ) | cos β. (5.7)On the other hand if we consider the map η : R → R given by η ( θ ) := | X (cos θ, sin θ ) | , since θ = 0 is a maximum point and X is smooth up to the boundary, then ψ ′ (0) = 0 and hence weget that X ( z ) · X v ( z ) = 0 . (5.8)Now consider the function ψ : R → R , given by ψ ( θ ) := X (cos θ, sin θ ) · ˆ p − | X (cos θ, sin θ ) | cos β .Since θ = 0 is a minimum point for ψ , and X is smooth up to the boundary, we get that ψ ′ (0) = 0,and taking into account of (5.8), we deduce that X v ( z ) · ˆ p = 0 . (5.9)Equations (5.8), (5.9) mean that X v ( z ) is orthogonal to both X ( z ) and ˆ p . Thus, for some λ ∈ R \ { } , it holds X v ( z ) = λ ˆ p ∧ X ( z ). Thanks to assumption (I), being X v ( z ) | X v ( z ) | the tangentversor to Γ at X ( z ) (we recall that, by Corollary 4.6, X has no boundary branch points) wehave that λ >
0, in particular X v ( z ) has the same direction and verse of ˆ p ∧ X ( z ). To provethis, we first observe that thanks to the definition of β -convexity ˆ p and X ( z ) must be linearlyindependent, so setting Π := Span { ˆ p , X ( z ) } we have that Π is a plane. Moreover, taking intoaccount of Assumption (I) and Remark 5.4, we have that P X induces a positive orientation on ∂ Ω. Hence, by Definition 5.3, since (
P X ) ′ (0) = X v ( z ) | X ( z ) | − X ( z ) · X v ( z ) | X ( z ) | X ( z ), we must have (cid:18) X v ( z ) | X ( z ) | ∧ X ( z ) (cid:19) · ˆ p = Det (cid:20) X v ( z ) | X ( z ) | , X ( z ) , ˆ p (cid:21) < . Hence, being X v ( z ) = λ ˆ p ∧ X ( z ), by the elementary properties of the determinant we getthat λ > -SURFACES IN CONES 16 Now let us consider the map | X | : B → R . Since X is an H-surface, and H satisfies (1.4) wehave that | X | is subharmonic. In fact, by elementary computations, we have ( | X | ) u = 2 X · X u ,( | X | ) uu = 2 X u · X u + 2 X · X uu and hence − ∆ | X | = − E − X · H ( X )( X u ∧ X v ) ≤ − E + 4 | X || H ( X ) || X u ∧ X v |≤ − (4 − c β ) E ≤ . In particular | X | − M is subharmonic and | X | − M ≤
0. Hence, by Hopf’s lemma, since | X | − M X would be a portion of a sphere, and hence | H ( X ) | ≡ √ M , whichcontradicts (1.4)), we get that X ( z ) · X u ( z ) > . (5.10)Now, let us observe that by construction and since X u · X v ≡ X u ( z ) ∈ Π. Wewant to understand where X u is located with respect to ˆ p and X ( z ). By construction the twovectors ˆ p and X ( z ) determine an angle of amplitude β . Let us denote by R the angular regionin Π generated by ˆ p and X ( z ), and by R its complementary in Π.We show that X u ( z ) R . In fact if X u ( z ) ∈ R , then, denoting by α ∈ ]0 , β ] the anglebetween X u ( z ) and X ( z ) ( α = 0 in view of Proposition 5.2, or by (5.7)) we have that β − α isthe angle between ˆ p and X u ( z ). Then, by dividing by | X u ( z ) | each side of (5.7), we get thatcos( β − α ) < cos( α ) cos( β ) , (5.11)and, by elementary trigonometry, we see that this last inequality is contradictory since both α and β are in ]0 , π/ X u ( z ) ∈ R and thanks to (5.10) X u ( z ) must also lie in the half-plane T := { p ∈ Π; p · X ( z ) > } . Thus, X u ( z ) ∈ R ∩ T , and let us consider the two subregions inwhich R ∩ T splits: R , , R , . R , is defined as the subset of R ∩ T such that ˆ p ∈ ∂R , .Arguing as in the previous case we see that X u ( z ) R , . In fact, if X u ( z ) ∈ R , , denotingby α ∈ ] β, π/
2[ the angle between X u ( z ) and X ( z ) we have that α − β is the angle between ˆ p and X u ( z ) and as before we havecos( α − β ) < cos( α ) cos( β ) , which is contradictory.At the end the only possibility is X u ( z ) ∈ R , . Now, since X v ( z ) = λ ˆ p ∧ X ( z ), by theelementary properties of the determinant we get that X ( z ) · ( X u ( z ) ∧ X v ( z )) = λ Det [ X u , X, X ∧ ˆ p ]and since X u ∈ R , we have that { X u , X, X ∧ ˆ p } is a positively oriented base of R . Hence weget that X ( z ) · X u ( z ) ∧ X v ( z ) >
0. Now, thanks to Proposition 5.2, we have that the function X · X u ∧ X v has a constant sign on ∂B . Hence N · X > ∂B and the proof is complete. (cid:3) From Proposition 5.1 and Proposition 5.6 we finally get the following:
Proposition 5.7.
Let Γ be a regular Jordan curve of class C ,α contained in C β \ { } and let X ∈ C ,α ( B, R ) be an H -surface spanning Γ . Suppose that Assumption (I) is satisfied. Then N · X > in B . Global invertibility of the radial projection
In this section we prove that under our assumptions the radial projection of an H -surface is ahomeomorphism, in particular it can be represented as a radial graph. At the end of this sectionwe prove Theorem 1.2. -SURFACES IN CONES 17 Theorem 6.1.
Let Γ be a regular Jordan curve of class C ,α contained in C β \ { } and let X bea stable H-surface of class C ,α ( B, R ) spanning Γ , with H satisfying (1.4) , (1.5) . Suppose thatAssumption (I) is satisfied. Then P X : B → Ω is a homeomorphism.Proof. The idea is to apply a classical result of global invertibility (see Theorem 2.10). We dividethe proof in four steps.
Step 1:
P X is a surjective map from B to Ω.Since X maps homeomorphically ∂B onto Γ, and Γ satisfies assumption (I) then P X mapshomeomorphically ∂B onto ∂ Ω (it is a composition of a homeomorphism and a continuousbijective map from a compact space into a Hausdorff space which is a homeomorphism too).Without loss of generality, assume that
P X ( B ) is contained in the upper hemisphere S + := S ∩ { z > } and let us denote by π : S \ P S → R the stereographic projection from the southpole P S = (0 , , − π ( P X ) maps homeomorphically ∂B onto π ( ∂ Ω) it follows that for deg ( π ( P X ) , q ) ≡ deg ( π ( P X ) , q ) ≡ −
1, where q ∈ π ( P X ( B )).In fact q π ( P X ( ∂B )) and by the basic properties of the degree (see for instance [5]) weknow that q deg ( π ( P X ) , q ) is constant in each connected component of R \ π ( P X ( ∂B )) (werecall that since π ( P X ( ∂B )) is a Jordan curve then R \ π ( P X ( ∂B )) has only two connectedcomponents), in particular it is constant for q ∈ π (Ω). Hence, being π ( P X ) : ∂B → ∂ Ω ahomeomorphism there are only two possibilities: deg ( π ( P X ) , q ) ≡ deg ( π ( P X ) , q ) ≡ − P X ∈ C ( B, R ), P ( B ) ⊂ Ω and being deg ( π ( P X ) , q ) = 0 for any q ∈ π (Ω), it follows that π ( P X )( B ) = π (Ω) (see [5]). Being π a diffeomorphism it follows that P X ( B ) = Ω. At the end, since P X maps ∂B onto ∂ Ω, we have
P X ( B ) = Ω. Hence P X is asurjective map from B to Ω. Step 2:
P X : B → Ω is locally invertible.We begin with the local invertibility of
P X : B → Ω. Being (
P X ) u = X u | X | − X · X u | X | X , byelementary computations we have( P X ) u ∧ ( P X ) v · P X = X u ∧ X v · X | X | . (6.1)Hence, thanks to Proposition 5.7, since N · X > B ′ of regular pointsof X it holds X u ∧ X v · X > , which, in view of (6.1), implies that ( P X ) u ( z ) and ( P X ) v ( z ) are linearly independent in B ′ .Thanks to a standard argument based on the the inverse function theorem it follows that P X is a local diffeomorphism except on a discrete set of critical points (given by the branch pointsof X ). Hence, from the standard properties of the degree (see for instance Theorem 2.9 in [5]),from Proposition 5.7 and since deg ( π ( P X ) , q ) ≡ ± q ∈ π (Ω), it follows that each regularvalue has exactly one pre-image. In fact let q ∈ π (Ω) be a regular value, then the set of pre-images of q is discrete and hence, being B compact, it is finite, and assuming for instance that deg ( π ( P X ) , q ) = 1 (see the proof of Step 1), by the index formula (see Theorem 2.9-(1) in [5])we get that 1 = deg ( π ( P X ) , q ) = X p ∈ [ π ( P X )] − ( q ) i ( π ( P X ) , p ) . (6.2)Now, being q a regular value we have that P X is local diffeomorphism at any p ∈ [ π ( P X )] − ( q ),and i ( π ( P X ) , p ) = ±
1. Thanks to Proposition 5.7 and (6.1) it follows that (
P X ) u ∧ ( P X ) v · P X > p ∈ [ π ( P X )] − ( q ). Hence, -SURFACES IN CONES 18 near each pre-image of q , P X has the same orientation, so it follows that i ( π ( P X ) , p ) ≡ i ( π ( P X ) , p ) ≡ −
1, for p ∈ [ π ( P X )] − ( q ). Thus, from (6.2), we deduce that i ( π ( P X ) , p ) ≡ deg ( π ( P X ) , q ) = X p ∈ [ π ( P X )] − ( q ) i ( π ( P X ) , p ) = k, where k ∈ N + is the cardinality of the set [ π ( P X )] − ( q ). Hence the only possibility is k = 1, i.e. q has only one pre-image.It remains to prove the local invertibility in the finite set of branch points. Let z be a branchpoint and assume by contradiction that P X is not invertible at z and set p := P X ( z ), then,for any neighborhood V of z we have that P X is not injective in V . Since branch points areisolated we can assume without loss of generality that V contains only z as branch point. Then,there exist z , z ∈ B , z = z such that P X ( z ) = P X ( z ) and necessarily one of them (forinstance z ) is not a branch point. It cannot happen that P X ( z ) is a regular value, since wehave proved that each regular value has exactly one pre-image. Hence P X ( z ) = P X ( z ) = p .By induction, repeating this argument we can construct a sequence of regular points ( z n ) ⊂ V , z n → z and such that z i = z j for any i = j . In particular S := P X − ( p ) is not finite. Now,up to an isometry we can assume that P X ( z ) = e and N ( z ) · e >
0. Let us denote byΠ e : R → R the projection of the first two coordinates. We observe that for any z ∈ S we haveΠ e ( X ( z )) = 0. On the other hand, arguing as in the proof of Theorem 1, Section 7.1 in [3] (inparticular, see (25)), since N ( z ) · e >
0, using the complex notation we can expand Π e ( X ( z ))near z as Π e ( X ( z )) = l (( z − z ) n +1 ) + o ( | z − z | n +1 ), where n = n ( z ) ∈ N is given by Theorem2.1 and l : C → C is the map associated to a nonsingular real matrix (see (24), Sect. 7.1, [3]).From this expansion we deduce that 0 must have a finite set of pre-images near z , and hencewe get a contradiction. Hence P X : B → Ω is locally invertible.Now we show that even
P X : B → Ω is locally invertible. In fact, as proved in Corollary 4.6,
P X has no boundary branch points, so, considering a suitable C -extension of P X , to someopen neighborhood V of ∂B we can assume that X has no branch points in V . Now, from (6.1),Proposition 5.7, we have X u ∧ X v · X > V which, in view of (6.1), implies that ( P X ) u ( z ) and ( P X ) v ( z ) are linearly independent in V .Hence, as before by an application of the inverse function theorem it follows that P X is a locallyinvertible for any z ∈ V and we are done. Step 3:
P X : B → Ω is proper.For any compact subset K ⊂ Ω we have that K is closed and being P X continuous we have(
P X ) − ( K ) is a closed subset of B . Being B compact it follows that ( P X ) − ( K ) is compact. Step 4:
Ω is simply connected.Thanks to an important result of differential geometry, known as Sch¨onflies’s Theorem or alsoJordan-Sch¨onflies’s Theorem (for the proof see for instance [16]) we know that the closure of thecomplement of the bounded region determined by a planar Jordan curve is homeomorphic to aclosed ball, in particular it is simply connected. Hence, taking the stereographic projection of Ω,since ∂ Ω is mapped onto a plane Jordan curve, we get that Ω is simply connected.From Step 1-Step 4, and being B arcwise connected we have that the hypotheses of Theorem2.10 are satisfied, so we get that P X is a homeomorphism. The proof is complete. (cid:3)
Remark 6.2.
An immediate consequence of the previous theorem is that the H-surface X canbe expressed as a radial graph. In fact, let ( P X ) − : Ω → B the inverse function of P X , and set -SURFACES IN CONES 19 F ( p ) := ( P X ) − ( p ) . Then, being X ( F ( p )) = P X ( F ( p )) | X ( F ( p )) | = p | X ( F ( p )) | , we get that X ( B ) = { q ∈ R ; q = λ ( p ) p, p ∈ Ω } , where λ : Ω → R + is the function defined by λ ( p ) := | X ( F ( p )) | .Proof of Theorem 1.2. Let X be the H -surface given by Theorem 1.1. Under our assumptionswe have that X ∈ C ,α ( B, R ) (see Proposition 2.3) and X is stable (see Remark 2.8). Froma remarkable result of Gulliver (see Theorem 8.1 and Theorem 8.2 in [6]) we have that X isfree of interior branch points and Corollary 4.6 excludes also boundary branch points. Hence,from the proof of Theorem 6.1 it follows that the radial projection of X is actually a globaldiffeomorphism. (cid:3) References [1] Abate, M., Tovena, F.:
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Plateau’s Problem and the Calculus of Variations , Princeton, N.J.: Princeton University Press(1989).[16] Thomassen, C.: The Jordan-Sch¨onfiles theorem and the classification of surfaces, Amer. Math. Monthly ,116–131 (1992).(Paolo Caldiroli) Dipartimento di Matematica, Universit`a di Torino, via Carlo Alberto, 10 – 10123Torino, Italy
E-mail address : [email protected] (Alessandro Iacopetti) Dipartimento di Matematica, Universit`a di Torino, via Carlo Alberto, 10 –10123 Torino, Italy
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