Uniqueness of stable closed non-smooth hypersurfaces with constant anisotropic mean curvature
aa r X i v : . [ m a t h . DG ] M a r Uniqueness of stable closed non-smooth hypersurfaces withconstant anisotropic mean curvature
Miyuki Koiso ∗ Abstract
We study a variational problem for piecewise-smooth hypersurfaces in the ( n + 1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is theintegral of an energy density that depends on the normal at each point over the con-sidered hypersurface. The minimizer of such an energy among all closed hypersurfacesenclosing the same ( n + 1)-dimensional volume is unique and it is (up to rescaling)so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this en-ergy for volume-preserving variations are not smooth in general. We prove that, if theanisotropic energy density function is of C and convex, then any closed stable equi-librium hypersurface is (up to rescaling) the Wulff shape. We also give fundamentaldefinitions, many examples, and generalizations of well-known concepts and formulaslike Steiner’s formula and Minkowski’s formula to the anisotropic case. Contents §
1. Introduction 2 §
2. Formulation of piecewise- C r weak immersion and anisotropic surface energy 6 §
3. Definition and characterizations of the Wulff shape 7 §
4. Convex integrand 9 §
5. Basic properties of the Cahn-Hoffman map 10 §
6. Curvatures and the regularities of the Wulff shape and the Cahn-Hoffman map 12 §
7. Examples 15 §
8. Anisotropic shape operator and anisotropic curvatures 20 §
9. First variation formula, anisotropic mean curvature, and anisotropic Gauss map 22 §
10. Anisotropic parallel hypersurface and Steiner-type formula 28 §
11. Minkowski-type formula 30 §
12. Proof of Theorem 1.1 33 ∗ This work was partially supported by JSPS KAKENHI Grant Number JP18H04487. 2010 Mathe-matics Subject Classification:49Q10, 53C45, 53C42. Key words and phrases: anisotropic mean curvature,anisotropic surface energy, Wulff shape, crystalline variational problem, Cahn-Hoffman vector field.
13. Proof of Proposition 9.1 36 § A. Computations for Example 7.5 39 § B. Proof of Lemma 8.1 (i) 40References 41
An anisotropic surface energy was introduced by J. W. Gibbs (1839-1903) in order to modelthe shape of small crystals ([28],[29]), which is the integral of an energy density that dependson the surface normal as follows. Let γ : S n → R ≥ be a non-negative continuous functionon the unit sphere S n = { ν ∈ R n +1 | k ν k = 1 } in the ( n + 1)-dimensional Euclideanspace R n +1 . Let X be a closed piecewise- C weakly immersed hypersurface in R n +1 (thedefinition of piecewise- C weakly immersed hypersurface will be given in § X will berepresented as a piecewise- C mapping X : M → R n +1 from an n -dimensional orientedconnected compact C ∞ manifold M into R n +1 . And the unit normal vector field ν along X is defined on M except a set with measure zero. Then, we can define the anisotropicenergy F γ ( X ) of X as F γ ( X ) := R M γ ( ν ) dA , where dA is the n -dimensional volume formof M induced by X . If γ ≡ F γ ( X ) is the usual n -dimensional volume of the hypersurface X . It is known that, if γ is positive on S n , for any positive number V >
0, among all closedhypersurfaces in R n +1 enclosing the same ( n + 1)-dimensional volume V , there exists aunique (up to translation in R n +1 ) minimizer W ( V ) of F γ ([24]). Here a closed hypersurfacemeans that the boundary (having tangent space almost everywhere) of a set of positiveLebesgue measure. The minimizer W ( V ) for V := ( n + 1) − R S n γ ( ν ) dS n is called theWulff shape (for γ ) (the standard definition of the Wulff shape will be given in § W γ or simply W . When γ ≡ W is the unit sphere S n . All W ( V ) arehomothetic to W . W is convex but not necessarily smooth. On the other hand, for anygiven convex set ˜ W having the origin of R n +1 inside, there exists a Lipschitz continuousfunction γ : S n → R > such that the boundary W := ∂ ˜ W of ˜ W is the Wulff shape for γ .However, such γ is not unique. The “smallest” γ is called the convex integrand for W (or,simply, convex) (for details and for another equivalent definition of the convexity, see § X of F γ for variations that preserve the enclosed ( n + 1)-dimensional volume (we will call such a variation a volume-preserving variation) has con-stant anisotropic mean curvature and satisfies a certain condition on its “edges” (Proposi-tion 9.2). Here the anisotropic mean curvature Λ of a piecewise- C weakly immersed hy-persurface X is defined at each regular point of X as (cf. [22], [15]) Λ := (1 /n )( − div M Dγ + nHγ ), where Dγ is the gradient of γ on S n and H is the mean curvature of X (see § γ ≡
1, then Λ = H holds.We call a piecewise- C equilibrium hypersurface X a CAMC (constant anisotropic meancurvature) hypersurface (see Definition 9.1 for details). A CAMC hypersurface is said tobe stable if the second variation of the energy F γ for any volume-preserving variation isnonnegative.In the special case where γ ≡
1, CAMC hypersurface is a CMC (constant meancurvature) hypersurface of C ω class (Corollary 9.1). For another special γ defined by γ ( ν , ν , ν ) = p | ν − ν − ν | , ( ν = ( ν , ν , ν ) ∈ S ), F γ ( X ) is the area of the surface X in R regarded as a surface in the Lorentz-Minkowski space R := ( R , dx + dx − dx ),and a CAMC surface is a CMC surface in R (Example 7.2). These examples suggest that F γ is a generalization of many important energy functionals for surfaces.A natural question is Question 1.1.
Is any closed CAMC hypersurface the Wulff shape (up to translation andhomothety)?
The answer to this uniqueness problem is not affirmative even in the case where γ ≡ X : M → R n +1 satisfies one of the following conditions (I)-(III), the image of X coincides with theWulff shape (up to translation and homothety).(I) X is an embedding, that is, X is an injective mapping.(II) X is stable.(III) n = 2 and the genus of M is 0, that is, M is homeomorphic to S .If we assume that the Wulff shape W γ is a smooth strictly convex hypersurface (that is, γ is of C and the n by n matrix A := D γ + γ · S n ,where D γ is the Hessian of γ on S n and 1 is the identity matrix of size n ), any closedCAMC hypersurface X is also smooth and the above expectation was already proved. Infact, if X satisfies at least one of (I)-(III), it is a homothety of the Wulff shape. This factwas proved, under the assumption that γ ∈ C ∞ ( S n ) and that A is positive definite, by thefollowing papers. (I): [1] for γ ≡
1, [9] for general γ . (II): [3] for γ ≡
1, [20] for general γ .(III): [11] for γ ≡
1, [16] and [2] for general γ .However, the situation is not the same for more general γ and/or W . Actually, thereexists a C ∞ function γ : S n → R > which is not a convex integrand such that there existclosed embedded CAMC hypersurfaces in R n +1 for γ each of which is not (any homothetyor translation of) the Wulff shape ([12]). Also, there exists a C ∞ function γ : S → R > which is not a convex integrand such that there exist closed embedded CAMC surfaces in R with genus zero for γ each of which is not (any homothety or translation of) the Wulffshape [12]. 3s for the uniqueness of stable closed CAMC hypersurfaces, we prove the followinguniqueness result in this paper. Theorem 1.1.
Assume that γ : S n → R > is of C and the convex integrand of its Wulffshape W . Then, the image of any closed stable piecewise- C CAMC weakly immersedhypersurface for γ whose r -th anisotropic mean curvature for γ is integrable for r = 1 , · · · , n is (up to translation and homothety) W . The assumption on the integrability of r -th anisotropic mean curvatures in Theorem1.1 is a natural one. In fact, r -th anisotropic mean curvatures are quantities that aredefined by ratios of principal curvatures of W and the normal curvatures of the consideredhypersurface (Definition 8.3), and each r -th anisotropic mean curvature of W is ( − r withrespect to the outward-pointing normal (Remark 8.1).We should remark that, although it is natural to study variational problems for anisotropicsurface energy for which equilibrium surfaces have singular points, it has not yet been donesufficiently well. As for planer curves, F. Morgan [18] proved that, if γ : S → R > iscontinuous and convex, then any closed equilibrium rectifiable curve for F γ in R with areaconstraint is (up to translation and homothety) a covering of the Wulff shape (see [19] foranother proof). About uniqueness of closed stable equilibria in R , B. Palmer [21] provedthe same conclusion as Theorem 1.1 but under the assumptions that γ : S → R > is of C and considered surfaces and the Wulff shape satisfy some extra assumptions (see Remark12.1 for details).We have another important remark about the assumption of the regularity of the energyfunctional in Theorem 1.1. If γ : S n → R > is a convex integrand, γ ∈ C , if and only if W γ is uniformly convex (F. Morgan [17])DThis result implies that, if we assume that γ isof C , then its Wulff shape W γ never has straight line segment (see § γ .As for the variational problem of F γ without volume constraint, under the assump-tion that γ ∈ C , there is no closed piecewise-smooth weakly immersed hypersurface in R n +1 that is a critical point of F γ (Proposition 9.5). This means that there is no closedpiecewise-smooth weakly immersed hypersurface in R n +1 whose anisotropic mean curvatureis constant zero.We give this article as the first article that gives a systematic research of the vari-ational problem of the anisotropic surface energy for piecewise-smooth weakly immersedhypersurfaces. Because of this reason we construct the theory from the beginning. Wewill not only give the proof of Theorem 1.1, we will give fundamental definitions and manyexamples, prove several important results on the curvatures of the Wulff shape and theCahn-Hoffman map, and derive important integral formulas for the critical points of our4ariational problem. This article is organized as follows.In §
2, we give the formulation of piecewise- C r weakly immersed hypersurface and thedefinition of its anisotropic energy.In §
3, we recall definitions of the Wulff shape and convexities of closed hypersurfaces. Wealso give the definition of the Cahn-Hoffman map ξ γ for γ (Definition 3.1) as the mapping ξ γ : S n → R n +1 defined as ξ γ ( ν ) = Dγ | ν + γ ( ν ) ν , here the tangent space T ν ( S n ) of S n at ν ∈ S n is naturally identified with the n -dimensional linear subspace of R n +1 . It willbe proved that, if γ : S n → R > is of C , then ξ γ is a C -(wave)front (Proposition 5.1).Moreover, it will be proved that, in this case the Wulff shape W γ is the uniquely-determinedconvex subset of the image ξ γ ( S n ) of ξ γ including the origin in R n +1 in its interior, andthat W γ = ξ γ ( S n ) if and only if γ is convex (Theorem 4.1).In §
4, we recall the definition of convex integrand and give an analytic representationof the Wulff shape.In §
5, we give some fundamental properties of the Cahn-Hoffman map.In §
6, we discuss curvatures and the regularities of the Wulff shape W γ and the Cahn-Hoffman map ξ γ . Especially, we prove that, if the energy density function γ : S n → R > isof C , then the principal curvatures of W γ and ξ γ are defined at each regular point and theynever vanish, and we give conditions so that they are unbounded (Theorem 6.1, Corollaries6.1, 6.2 ). We also prove that, even when some of the principal curvatures of them areunbounded, the integral of the mean curvature is finite (Theorem 6.2).In §
7, we give several examples of the energy density functions and corresponding Wulffshapes and Cahn-Hoffman maps.In §
8, first we define the Cahn-Hoffman field ˜ ξ γ := ξ γ ◦ ν along X (Definition 8.1),which is a generalization of the Gauss map and is called also the anisotropic Gauss map of X (for γ ). Then we give the definition of the anisotropic shape operator (Definition 8.2)and its representation in terms of local coordinates (Lemma 8.2). And the definitionsof anisotropic curvatures including the anisotropic mean curvature (Definition 8.3) aregiven. We also give the representation of the anisotropic mean curvature in terms of theprincipal curvatures of the Cahn-Hoffman map and the corresponding normal curvaturesof the considered hypersurface (Remark 8.2).In §
9, we derive the Euler-Lagrange equations for our variational problem (Proposition9.2). In general the anisotropic Gauss map ˜ ξ γ = ξ γ ◦ ν is not well-defined as a single-valuedmapping because the Gauss map ν is not well-defined on the singular points of X . However,if γ is convex and X is CAMC, then strikingly ˜ ξ γ is well-defined over the whole hypersurface(Theorem 9.1).In §
10 and §
11, we give the definition of anisotropic parallel hypersurface (Definition10.1) and prove a Steiner-type integral formula (Theorem 10.1) and a Minkowski-type5ntegral formula (Theorem 11.1). They are generalizations of important and useful concept(parallel hypersurfaces) and integral formulas (Steiner formula and Minkowski formula) inthe isotropic case to the anisotropic case. We will use them to prove Theorem 1.1 ( § § § § §
13 and in the appendices. C r weakly immersed hypersur-face and anisotropic surface energy First we formulate a piecewise- C r weak immersion for r ∈ N . Let M = ∪ ki =1 M i be an n -dimensional oriented compact connected C ∞ manifold, where each M i is an n -dimensionalconnected compact submanifold of M with piecewise- C ∞ boundary, and M i ∩ M j = ∂M i ∩ ∂M j , ( i, j ∈ { , · · · , k } , i = j ). We call a map X : M → R n +1 a piecewise- C r weakimmersion (or a piecewise- C r weakly immersed hypersurface) if X satisfies the followingconditions (A1), (A2), and (A3) ( i ∈ { , · · · , k } ).(A1) X is continuous, and each X i := X | M i : M i → R n +1 is of C r .(A2) If we denote by M oi the interior of M i , X | M oi is a C r -immersion.(A3) The unit normal vector field ν i : M oi → S n along X i | M oi can be extended to a C r − -mapping ν i : M i → S n . Here, if ( u , · · · , u n ) is a local coordinate system in M i , then { ν i , ∂/∂u , · · · , ∂/∂u n } gives the canonical orientation in R n +1 .The image X ( M ) of a piecewise- C r weak immersion X : M → R n +1 is also called apiecewise- C r weakly immersed hypersurface. Denote by S ( X ) the set of all singular pointsof X , here a singular point of X is a point in M at which X is not an immersion. Lemma 2.1.
Assume that X : M = ∪ ki =1 M i → R n +1 satisfies the above (A1), (A2), and(A3). Then, the n -dimensional Hausdorff measure of X ( ∂M i ) is zero for any i ∈ { , · · · , k } .Proof . Let ( u , · · · , u n − ) be local coordinates on ∂M i . Set g ij = h X u i , X u j i , i, j = 1 , · · · , n − . Then, the ( n − ∂M i induced by X is(det( g ij )) / du ∧ · · · ∧ du n − . From this and the compactness of ∂M i , the ( n − ∂M i is finite.This proves the desired result. ✷ Next we define the anisotropic energy of a piecewise- C weak immersion X : M → R n +1 .Assume that γ : S n → R ≥ is a nonnegative continuous function. Let ν : M \ S ( X ) → S n be the unit normal vector field along X | M \ S ( X ) . Since the n -dimensional Hausdorff measure6f X ( S ( X )) is zero, it is reasonable to define the anisotropic energy F γ ( X ) of X as F γ ( X ) := Z M γ ( ν ) dA. If γ ≡ F γ ( X ) is the usual n -dimensional volume of the hypersurface X (that is the n -dimensional volume of M with the metric induced by X ).It is often useful to consider the homogeneous extension γ : R n +1 → R ≥ of γ , which isdefined as follows. γ ( rX ) := rγ ( X ) , ∀ X ∈ S n , ∀ r ≥ . (1)If γ is a convex function on R n +1 (that is, γ ( v + v ) ≤ γ ( v ) + γ ( v ) holds for all v , v ∈ R n +1 ), γ is called a convex integrand or simply convex (see § In this section, we assume that γ : S n → R > is a positive continuous function. We givethe definition of the Wulff shape and some of its characterizations which we will need in thefollowing sections (see [24] for details). A positive continuous function on S n is sometimescalled an integrand.The boundary W γ of the convex set ˜ W [ γ ] := ∩ ν ∈ S n (cid:8) X ∈ R n +1 |h X, ν i ≤ γ ( ν ) (cid:9) is calledthe Wulff shape for γ , where h , i means the standard inner product in R n +1 . ( ˜ W [ γ ] itselfis often called the Wulff shape.) In the special case where γ ≡ W γ coincides with S n .If the homogeneous extension γ : R n +1 → R ≥ of γ is convex and satisfies γ ( − X ) = γ ( X ), ( ∀ X ∈ R n +1 ), γ defines a norm of R n +1 . The Wulff shape W γ coincides with theunit sphere { Y ∈ R n +1 | γ ∗ ( Y ) = 1 } of the dual norm γ ∗ of γ which is defined as γ ∗ ( Y ) = sup { Y · Z | γ ( Z ) ≤ } , ∀ Y ∈ R n +1 .W γ is not smooth in general. W γ is smooth and strictly convex (that is, each principalcurvature of W γ with respect to the inward-pointing normal is positive at each point of W γ ) if and only if γ is of C and the n by n matrix D γ + γ · S n , where D γ is the Hessian of γ on S n and 1 is the identity matrix of size n (cf.Corollary 6.2). When γ satisfies such a convexity condition, the functional F γ is sometimescalled a (constant coefficient) parametric elliptic functional ([5], [6]). Definition 3.1.
Assume that γ : S n → R ≥ is of C . We call the continuous map ξ :7 n → R n +1 defined as ξ ( ν ) := ξ γ ( ν ) := Dγ | ν + γ ( ν ) ν, ν ∈ S n (2) the Cahn-Hoffman map (for γ ). Here the tangent space T ν ( S n ) of S n at ν ∈ S n is naturallyidentified with the n -dimensional linear subspace of R n +1 . Remark 3.1.
It is easy to show that the Cahn-Hoffman map ξ γ : S n → R n +1 is representedby using γ as ξ γ ( ν ) = Dγ | ν , ν ∈ S n , (3) where D is the gradient in R n +1 . If γ : S n → R > is of C , the Wulff shape W γ is a subset of the image ˆ W γ := ξ γ ( S n ) of ξ γ , and ˆ W γ = W γ holds if and only if γ is convex (Theorem 4.1). Definition 3.2.
Assume that S is a closed hypersurface in R n +1 that is the boundary of abounded open set Ω. Denote by Ω the closure of Ω, that is, Ω = Ω ∪ S .(i) S is said to be strictly convex if, for any straight line segment P Q connecting twodistinct points P and Q in S , P Q ⊂ Ω and
P Q ∩ S = { P, Q } hold.(ii) S is said to be convex if, for any straight line segment P Q connecting two points P and Q in S , P Q ⊂ Ω holds.(iii) (cf. [23]) Assume that S is convex. The support function of S is a mapping σ S : S n → R which is defined as follows. For ν ∈ S n and a ∈ R , define a closed half spaceΠ( ν, a ) as Π( ν, a ) := (cid:8) X ∈ R n +1 | h X, ν i ≤ a (cid:9) . Then, set σ S ( ν ) := min { a ∈ R | Π( ν, a ) ⊃ S } . Now assume that γ is of C and convex. Then, for any ν ∈ S n , the outward-pointingunit normal to W γ at the point ξ ( ν ) is well-defined and it coincides with ν (Proposition5.1). Hence γ is the support function of W γ , because h ξ ( ν ) , ν i = h Dγ | ν + γ ( ν ) ν, ν i = γ ( ν )holds.Conversely, consider an arbitrary piecewise- C strictly convex closed hypersurface W in R n +1 . Assume that the origin of R n +1 is contained in the open domain bounded by W .Denote by γ the support function of W . Then, W is the Wulff shape for γ (cf. [24]).Therefore, there is a one-to-one correspondence between the following two sets A and B . A = { W ⊂ R n +1 | W is a piecewise- C strictly convex closed hypersurface, andthe origin of R n +1 is contained in the open domain bounded by W. } , (4) B = { γ : S n → R > | γ is of C and convex . } . (5)8he correspondence is given by the bijection F : A → B which maps W ∈ A to the supportfunction of W . The inverse mapping F − : B → A maps γ ∈ B to its Wulff shape W γ .An integrand γ is said to be crystalline if its Wulff sape W γ is a polyhedral hypersurface([24]). In this section, we prove that the Cahn-Hoffman map ξ γ (Definition 3.1) gives a represen-tation of the Wulff shape W γ if γ is of C and convex (Theorem 4.1). Definition 4.1.
For any nonnegative continuous function γ : S n → R ≥ , the set { γ ( ν ) ν ; ν ∈ S n } is called the Wulff plot for γ (or the radial plot of γ ). Definition 4.2 ([24]) . A positive continuous function γ : S n → R > is called a convexintegrand if the Wulff plot of the function /γ : S n → R > , (1 /γ )( ν ) := γ ( ν ) − , ∀ ν ∈ S n , is convex. Remark 4.1 ([24]) . (i) For any continuous γ : S n → R > , there exists a unique convexintegrand ˜ γ such that W γ = W ˜ γ holds.(ii) ˜ γ is the smallest integrand among all integrands having the same Wulff shape, thatis ˜ γ ( ν ) = min { f ( ν ) | f ∈ C ( S n , R > ) , W f = W ˜ γ } , ∀ ν ∈ S n holds. Recall that γ : R n +1 → R ≥ is the homogeneous extension of γ : S n → R ≥ ( § Remark 4.2 (cf. § . Assume that γ : S n → R > is continuous. Then γ is convexif and only if γ is a convex function.Although maybe the following result has been already proved somewhere, we will giveits proof for completeness. Theorem 4.1.
Assume that γ : S n → R > is of C .(i) The Wulff shape W γ for γ is a subset of the image ˆ W γ := ξ γ ( S n ) of the Cahn-Hoffmanmap ξ γ : S n → R n +1 .(ii) The following (a), (b) and (c) are equivalent.(a) γ is a convex integrand.(b) W γ = ˆ W γ holds. c) D γ + γ · is positive-semidefinite, that is, the eigenvalues are all nonnegativeon the tangent space at each point in S n .(iii) If γ is uniformly convex, that is, the eigenvalues of D γ + γ · are all positive onthe tangent space at each point in S n , then ξ γ : S n → R n +1 is an embedding onto W γ . Inthis case, for each ν ∈ S n , ν coincides with the outward-pointing unit normal of W γ at thepoint ξ γ ( ν ) , and γ is the support function of W γ , that is, γ ( ν ) = h ξ γ ( ν ) , ν i holds.Proof. First we prove (i). The support function of W γ in the sence of Proposition 5.2coincides with γ almost everywhere on W γ . On the other hand, the support function of ξ γ is exactly γ : S n → R > . Since a support function determines the hypersurface uniquely, W γ ⊂ ξ γ ( S n ) holds.Now we prove (ii). First we prove that (a) and (c) are equivalent. Note that γ is convex(that is, γ ( v + v ) ≤ γ ( v ) + γ ( v ) holds for all v , v ∈ R n +1 ) if and only if the Hessian Hess ( γ ) of γ is positive-semidefinite. It is sufficient to prove that D γ + γ · ν = (0 , · · · , , x , · · · , x n ) as local coordinate of S n around ν , where ( x , · · · , x n +1 ) is the canonical coordinate in R n +1 . Then, D γ = n X i,j =1 ∂ γ∂x i ∂x j − n X k =1 Γ kij ∂γ∂x k ! dx i ⊗ dx j , where Γ kij is the Christoffel symbols. Since ν gives the normal direction of ˆ W at ξ ( ν )(Proposition 5.1), and since γ is homogeneous, we can show that D γ + γ · Hess ( γ ) (cid:12)(cid:12) S n holds. This implies that (a) and (c) are equivalent.Next we prove that (a) and (b) are equivalent. γ is convex if and only if γ is thesupport function of W γ ([24]). On the other hand, γ is the support function of ξ γ in thesence of Proposition 5.2. The property that the support functions of W γ and ξ γ coincide isequivalent to the property that W γ and ξ γ ( S n ) coincide.Lastly we prove (iii). The first statement follows from the fact that dξ γ = D γ + γ · ✷ In Theorem 4.1, we observed that, if γ : S n → R > is of C and convex, then the Cahn-Hoffman map ξ γ gives a parametrization of the Wulff shape W γ . In this section we alsoassume that γ is of C , but we do not assume that γ is convex, and we discuss geometry of ξ γ , especially its singular points and its curvatures.First we give some fundamental properties of ξ γ .10 roposition 5.1. Assume that γ : S n → R > is of C . Then, the Cahn-Hoffman map ξ := ξ γ satisfies the following (i) and (ii), hence ξ is a C -(wave)front.(i) h ( dξ ) ν ( u ) , ν i = 0 , ∀ ν ∈ S n , ∀ u ∈ T ν S n . (6) (ii) The mapping ( ξ, id S n ) : S n → R n +1 × S n , ( ξ, id S n )( ν ) := ( ξ ( ν ) , ν ) (7) is a C -immersion. We will give the proof of Proposition 5.1 at the end of this section for completeness.Proposition 5.1 (i) yields the following Definition 5.1 and Corollary 5.1.
Definition 5.1.
Assume that γ : S n → R > is of C . Then, at any point ν ∈ S n we callthe hyperplane perpendicular to ν the tangent hyperplane of ξ γ at ν (or at ξ γ ( ν )) . Corollary 5.1.
Assume that γ : S n → R > is of C . Then, at any point ν ∈ S n where ξ γ is an immersion, ν itself gives a unit normal to ξ γ . The following proposition is an immediate consequence of h ξ γ ( ν ) , ν i = γ ( ν ), which givesan important relation between γ and its Cahn-Hoffman map ξ γ . Proposition 5.2.
Assume that γ : S n → R > is of C . Then, γ is the support function of ξ γ , that is, γ ( ν ) is the distance between the origin of R n +1 and the tangent hyperplane of ξ γ at the point ξ γ ( ν ) .Proof of Proposition 5.1. (ii) is obvious because id S n : S n → S n is an immersion.We will prove (i). Let ( V, ϕ ) = ( V, ( y , · · · , y n )) be a local coordinate neighborhood of S n . Represent ξ = ( ξ , · · · , ξ n +1 ). Then( dξ ) ν (cid:16) ∂∂y j (cid:17) ν = ∂ξ ◦ ϕ − ∂y j . (8)Set V ′ := ϕ ( V ) and ψ := ϕ − : V ′ → V. The standard Riemannian metric ds = n X i,j =1 g ij dy i dy j in V is given by g ij := h ψ y i , ψ y j i . Set ( g ij ) := ( g ij ) − , and using the identification ∂∂y j = ψ y j , the gradient Dγ of γ is given as Dγ = n X i,j =1 g ij γ y i ψ y j , γ y i := ∂ ( γ ◦ ϕ − ) ∂y i . ξ = Dγ + γ ( ν ) ν = n X i,j =1 g ij γ y i ψ y j + γ ( ν ) ν, and hence ∂ξ∂y k = X i,j ( g ij γ y i ψ y j ) y k + ( γ ( ν )) y k ν + γ ( ν ) ν y k = X i,j ( g ij γ y i ) y k ψ y j + X i,j g ij γ y i ψ y j y k + ( γ ( ν )) y k ν + γ ( ν ) ν y k =: I + II + III + IV. (9)It is clear that h I, ν i = 0 , h IV, ν i = 0 (10)holds. We will compute the normal component of II. Differentiate h ν, ψ y j i = 0 to get h ν y k , ψ y j i + h ν, ψ y j y k i = 0 . Hence, h ν, ψ y j y k i = −h ν y k , ψ y j i = h kj , here P i,j h ij dy i dy j is the second fundamental form on S n . This gives B := h X i,j g ij γ y i ψ y j y k , ν i = X i,j g ij h kj γ y i . (11)In order to prove (6), let ν ∗ be the antipodal point of ν , and let π : S n \ { ν ∗ } → R n be thestereographic projection from ν ∗ . Then, at ν ,( g ij ) = ( δ ij ) , ( h ij ) = − ( δ ij ) (12)holds. Therefore we have B = − X i,j δ ij δ kj γ y i = − γ y k . (13)By (8), (9), (10), and (13), we obtain h ( dξ ) ν ( u ) , ν i = 0 , ∀ u ∈ T ν S n , which proves (i). ✷ The curvatures and the regularities of the Wulff shape and the Cahn-Hoffman map dependon the anisotropic energy density function γ , which is the subject of this section. We willexplain why we assume that γ is of C , not of C in our main results. Also we will observethat the assumption γ ∈ C is too strong in a certain sence.12irst we recall the following result. Remark 6.1.
Assume that γ : S n → R > is a convex integrand.(i) (F. Morgan [17]) γ ∈ C , if and only if W γ is uniformly convexD(ii) (H. Han and T. Nishimura [7]) γ ∈ C if and only if W γ is strictly convexDThis result implies that if a convex integrand γ : S n → R > is of C , then W γ does notinclude any straight line segment. In this case, γ is not crystalline, that is, the Wulff sape W γ is not a polyhedral hypersurface. In this sence, γ ∈ C is a too strong assumption.Actually, if a convex integrand γ : S n → R > is differentiable at all points in S n , then it isof C ([23]). Therefore, we have: Remark 6.2.
Assume that γ : S n → R > is a convex integrand. Then, W γ includes astraight line segment ℓ if and only if γ is not differentiable at a certain point ν ∈ S n . Moreprecisely, ν is normal to ℓ and γ is not partially differentiable at ν in the direction of ℓ .Next we consider the case where γ is of C . Theorem 6.1. If γ : S n → R > is of C , then the following (i) and (ii) hold.(i) The principal curvatures at any regular point of the Cahn-Hoffman map ξ γ nevervanish.(ii) For any singular point ν ∈ S n of ξ γ , and for any smooth one-parameter family ν t ∈ S n with lim t →∞ ν t = ν of regular points of ξ γ with principal curvatures µ ( t ) , · · · , µ n ( t ) ,the limit lim t →∞ | µ i ( t ) | exists and it is either ∞ or a nonzero real value, ( i = 1 , · · · , n ).Moreover, near any singular point ν of ξ γ , at least one of the principal curvatures of ξ γ isunboundedDProof. (i) Note that the operator D γ + γ · S n . Recall that ξ γ : S n → R n +1 is defined by ξ ( ν ) = Dγ | ν + γ ( ν ) ν . Since, at any regular point ν ∈ S n of ξ γ , ν itself givesa unit normal to ξ γ (Corollary 5.1), ξ γ gives the inverse of a unit normal vector field along ξ γ itself near ν . Hence, A := dξ γ = D γ + γ · ξ γ , which implies that the eigenvalues ρ , · · · , ρ n of A are thenegatives of the reciprocals of the principal curvatures µ , · · · , µ n of ξ γ with respect to theunit normal ν . This means that µ j = 1 /ρ j = 0 for all j ∈ { , · · · , n } at any point in S n .(ii) If ν ∈ S n is a singular point of ξ γ , the matrix A has at least one zero eigenvalue.This fact with the observation above gives the desired result. ✷ Example 7.7 gives a good example for Theorem 6.1 (ii).Recall that, if γ : S n → R > is of C and convex, then the Cahn-Hoffman map ξ γ givesa parametrization of the Wulff shape W γ (Theorem 4.1). This fact with Theorem 6.1 (ii)gives the following results. 13 orollary 6.1. Assume that γ : S n → R > is convex. If γ is of C , then near any singularpoint of W γ , at least one of the principal curvatures of W γ is unboundedD Examples 7.5, 7.6 give good examples for Corollary 6.1.The following known result is an immediate consequence of Theorem 6.1 (i).
Corollary 6.2.
For γ ∈ C ( S n , R > ) , the following (i) and (ii) are equivalent.(i) W γ is a closed strictly-convex smooth hypersurface, that is, all of the principal cur-vatures of W are positive for the inward-pointing unit normal.(iii) D γ + γ · is positive-definite, that is, the eigenvalues are all positive, on the tangentspace at each point in S n . Even when some of the principal curvatures of ξ γ are unbounded, the integral of themean curvature of ξ γ is finite as follows. Theorem 6.2.
Assume that γ : S n → R > is of C . Then the mean curvature H of theCahn-Hoffman map ξ γ is defined on S n \ S ( ξ γ ) , and its improper integral Z S n \ S ( ξ γ ) H dA converges.Proof.
Take any point ν ∈ S n . Because dξ γ = D γ + γ · { e , · · · , e n } on S n such that ( D γ + γ · e i = ρ i e i holds at ν , where ρ i are the eigenvalues of dξ γ . Because of Proposition 5.1 (i), the basis { e , · · · , e n } at ν alsoserves as an orthogonal basis for the tangent hyperplane of ξ γ at ν . If ν is a regular pointof ξ γ , as the proof of Theorem 6.1 (i), ρ , · · · , ρ n are the negatives of the reciprocals of theprincipal curvatures µ , · · · , µ n of ξ γ with respect to the unit normal ν . Hence dξ γ = ( D γ + γ ·
1) = /µ · · ·
00 1 /µ · · ·
00 0 · · · · · · · · · · · · · · · · ·
00 0 · · · /µ n . holds. Therefore we have nH dA = ( µ + · · · + µ n ) 1 | µ · · · µ n | du ∧ · · · ∧ du n , (14)where ( u , · · · , u n ) is the corresponding local coordinate in S n . Here, | µ + · · · + µ n | | µ · · · µ n | ≤ | µ || µ · · · µ n | + · · · + | µ n || µ · · · µ n | = | ρ · · · ρ n | + · · · + | ρ · · · ρ n − | (15)holds. Since γ is of C , the right hand side of (15) is bounded, which implies the desiredresult. ✷ Examples
In this section, we give several examples of integrands γ : S n → R > with various regulari-ties, and we show their Wulff shapes and Cahn-Hoffmann maps. Sometimes we will assume n = 1 or n = 2 for simplicity. However, it is easy to generalize them to higher dimensionalexamples (for example, by rotation). Below we use the notation ν = ( ν , · · · , ν n +1 ) for ν ∈ S n . Example 7.1 (cf.[22]. γ ∈ C ∞ , uniformly convex) . Set n = 2. Let a , a , a be positiveconstants. Set γ ( ν ) := q a ν + a ν + a ν . Then, γ ∈ C ∞ and it is uniformly convex. W γ is the ellipsoid x a + x a + x a = 1 . Since, by the transformation x ′ = x /a , x ′ = x /a , x ′ = x /a , the functional F γ = Z q a ν + a ν + a ν dA becomes a a a times the usual area, an immersion X = ( x , x , x ) has constant anisotropicmean curvature if and only if ( x ′ , x ′ , x ′ ) has constant mean curvature. Example 7.2 ([10]. γ : S n → R ≥ , γ ∈ C , γ / ∈ C , γ is not convex) . Set n = 2. Define γ : S → R ≥ as γ ( ν ) := q | ν − ν − ν | . Then, F γ ( X ) is the area of the surface X regarded as a surface in the Lorentz-Minkowskispace R := ( R , dx + dx − dx ). Hence, a CAMC surface is a CMC surface in R . For agraph z = f ( x, y ), Λ and the mean curvature H L as a surface in R satisfyΛ = H L = (1 / | − f x − f y | − / h (1 − f y ) f xx + 2 f x f y f xy + (1 − f x ) f yy i . Hence, the equation Λ = constant is(i) elliptic on “space-like parts” where 1 − f x − f y > − f x − f y < R . From the left, thespace-like catenoid, the space-like Riemann-type maximal surface, the time-like catenoid,and the time-like Riemann-type minimal surface. Example 7.3 (cf.[24]. γ ∈ C and convex, but γ / ∈ C ) . Set γ ( ν ) = n +1 X i =1 | ν i | . Then, γ ∈ C and convex, but γ / ∈ C . W γ is the cube { x = ( x , · · · , x n +1 ) ∈ R n +1 | max {| x | , · · · , | x n +1 |} =1 } . We show a picture (Figure 2) for the special case where n = 1. The Wulff shape W γ is the square drawn with straight line segments, and the Wulff plot is the dotted curves(Figure 2, left). γ is not differentiable at the four points ( ± , , (0 , ± ξ γ : S \ { ( ± , , (0 , ± } → R is the set of the four points { (1 , , ( − , , (1 , − , ( − , − } (Figure 2, right). - - - - ææææ - - - - Figure 2: Left: The Wulff shape (solid curves) and the Wulff plot (dotted curves) of Example7.3. Right: The image of the Cahn-Hoffman map ξ γ : S \ { ( ± , , (0 , ± } → R . Example 7.4 ( γ ∈ C and convex, but γ / ∈ C ) . Let r , h be positive constants. Define γ : S n → R > as follows. γ ( ν ) = r q ν + · · · + ν n + h | ν n +1 | , ν = ν , · · · , ν n +1 ∈ S n . (16)Then, γ ∈ C ( S n ) and convex, but γ is not differentiable at any point in the following set S ( γ ). S ( γ ) := { ( ν , · · · , ν n +1 ) ∈ S n | ν n +1 = 0 , ± } . (17)The Wulff shape W γ is the cylinder C with radius r and height 2 h covered with two flat16iscs with radius r , that is W γ = C ∪ Γ + ∪ Γ − , (18)where C = { ( x , · · · , x n +1 ) ∈ R n +1 | q x + · · · + x n = r, x n +1 ∈ [ − h, h ] } , (19)Γ ± = { ( x , · · · , x n , ± h ) ∈ R n +1 | q x + · · · + x n ≤ r } . (20)The Cahn-Hoffman map ξ : S n \ S ( γ ) → R n +1 is given as follows. ξ ( p cos θ, sin θ ) = ( rp, sgn(sin θ ) · h ) , p ∈ S n − , θ ∈ ( − π/ , ∪ (0 , π/ , (21)that is, ξ maps a quater circle to one point, and the image ξ ( S n \ S ( γ )) is the union of twocircles with radius r with height ± h as follows. ξ ( S n \ S ( γ )) = { ( x , · · · , x n , ± h ) ∈ R n +1 | q x + · · · + x n = r } (22)(Figure 3).Here we give a comment about the relationship between the singular points of γ andthe flat faces of W γ . Each singular point (0 , · · · , , ±
1) corresponds to the n -dimensionaldisc Γ ± , and each singular point ( p,
0) := ( ν , · · · , ν n ,
0) corresponds to the straight linesegment { ( rp, x n +1 ) ∈ R n +1 | − h ≤ x n +1 ≤ h } .Figure 3: The Wulff shape (a cylinder covered with two flat discs) and the image of theCahn-Hoffman map (two thick circles at the top and at the bottom) of Example 7.4 Example 7.5 ( γ ∈ C ∞ and convex, but not uniformly convex) . Set n = 1. For m ∈ N ,define γ m ( ν ) := ( ν m + ν m ) / (2 m ) . Then γ is of C ∞ and convex. γ is uniformly convex for m = 1, and it is not uniformly convex for m ≥ - - - - - - - Figure 4: Left: W γ for m = 1, right: W γ for m = 2, for γ m ( ν ) := ( ν m + ν m ) / (2 m ) inExample 7.5.By computations (see the appendix § A), we obtain the following results. Set ν =(cos θ, sin θ ). Then, the Cahn-Hoffman map ξ m for γ m is represented as follows. ξ m (cos θ, sin θ ) := ( f m ( θ ) , g m ( θ ))= (cos m θ + sin m θ ) (1 / (2 m )) − (cos m − θ, sin m − θ ) . (23)Also we have A m := D γ m + γ m ·
1= (2 m −
1) cos m − θ sin m − θ (cos m θ + sin m θ ) (1 / (2 m )) − . (24)Hence,(i) If m ≥ A m = 0 at θ = (1 / ℓπ , ( ℓ ∈ Z ).(ii) A m is positive-definite on S \ { (cos θ, sin θ ) | θ = (1 / ℓπ, ( ℓ ∈ Z ) } .The curvature κ m of ξ m with respect to the outward-pointing normal ν is represented as κ m = − f ′ m g ′′ m + f ′′ m g ′ m (( f ′ m ) + ( g ′ m ) ) / = − m − − m +2 θ sin − m +2 θ (cos m θ + sin m θ ) − m . (25)Hence, for any ℓ ∈ Z and m ≥
2, lim θ → ℓ π κ m ( θ ) = −∞ (26)holds. However, since κ m ds = − dθ (27)holds, Z π κ m ds = − Z π dθ = − π. (28)However, (28) is trivial because ξ m : S → R is a front with ν ∈ S . Example 7.6 ( γ ∈ C and convex, but γ / ∈ C ) . Set n = 1. Let C r be the convexclosed curve passing the four points (1 , , (1 , − , ( − , , ( − , −
1) which is the union offour circular arcs with radius r ≥ √ C √ is the circle with center at (0 , r = ∞ . We18ill write any point ν ∈ S as ν = (cos θ, sin θ ). The support function γ r : S → R > of C r is given by the symmetric extension of the following equation. γ r ( θ ) = cos θ + sin θ, π/ ≤ θ ≤ ( π/ − sin − (1 /r ) ,r + (1 − √ r −
1) sin θ, ( π/ − sin − (1 /r ) < ( π/
2) + sin − (1 /r ) , | cos θ | + sin θ, ( π/
2) + sin − (1 /r ) ≤ θ ≤ (3 / π. (29)Note γ ∞ ( θ ) := lim r →∞ γ r ( θ ) = | cos θ | + | sin θ | , ≤ θ ≤ π, which coincides with γ in Example 7.3.When √ < r < ∞ , by computation we can show that γ r ∈ C and convex, but γ r / ∈ C .If √ ≤ r < ∞ , both ξ γ r ( S ) and W γ r coincide with C r . When, r = ∞ , The imageof the Cahn-Hoffman map ξ γ ∞ : S \ { ( ± , , (0 , ± } → R is the set of the four points { (1 , , ( − , , (1 , − , ( − , − } (Figure 5, right). - - - - ææææ - - - - Figure 5: Left: The closed curves C r in Example 7.6. The solid curve: r = 2Cthe dashedcurve: r = 3, the dotted curve: r = + ∞ . When √ ≤ r < ∞ , C r = W γ r = ξ γ r ( S ). Right:The image of the Cahn-Hoffman map ξ γ ∞ : S \ { ( ± , , (0 , ± } → R . Example 7.7 ( γ ∈ C ∞ , and γ is not convex) . Set n = 1. Define γ as γ ( ν ) = 4 ν − ν + 2.Then, γ ∈ C ∞ , and γ is not convex. The whole of the closed curve with self-intersection inFigure 6 is the image ˆ W γ := ξ γ ( S ) of the Cahn-Hoffman map ξ γ , while the closed convexsolid curve that is a proper subset of ˆ W γ is the Wulff shape W γ . By computation, we obtain γ (cos θ, sin θ ) = 4 cos θ − θ + 2 =: γ ( θ ) , (30) ξ γ ( θ ) = (8 cos θ sin θ + 4 cos θ + 2 cos θ − , − θ sin θ + 2 sin θ ) =: ( f ( θ ) , g ( θ )) , (31) A := dξ γ = 2( −
16 cos θ + 12 cos θ + 1)( − sin θ, cos θ ) =: a ( θ )( − sin θ, cos θ ) , (32) κ γ ( θ ) := − f ′ g ′′ + f ′′ g ′ (( f ′ ) + ( g ′ ) ) / = − | −
16 cos θ + 12 cos θ + 1) | = − | a ( θ ) | . (33)Hence, for any ρ ∈ a − (0), lim θ ( / ∈ a − (0)) → ρ κ γ ( θ ) = −∞ . (34) a − (0) ⊂ S is the set of exactly six points, and they correspond to the singular points of ξ γ (Figure 6). 19 - - - - Figure 6: The image of the Cahn-Hoffman map ξ γ for γ in Example 7.7. The six verticesare the image of the singular points of ξ γ . The closed convex solid curve is the Wulff shape W γ . Assume that γ : S n → R > is of C . Let M be an oriented compact connected n -dimensional C ∞ manifold with smooth boundary ∂M . As in §
9, we assume that a map X : M → R n +1 satisfies the conditions (A1), (A2), and (A3) in § r = 2, X i = X , M i = M , and ν i = ν . Definition 8.1.
The Cahn-Hoffman field along X (for γ ) is defined as˜ ξ := ˜ ξ γ := ξ ◦ ν : M → R n +1 , (35)where ξ ( ν ) := ξ γ ( ν ) = Dγ + γ ( ν ) ν , ( ν ∈ S n ), is the Cahn-Hoffman map for γ . ˜ ξ γ is calledalso the anisotropic Gauss map of X (for γ ). Definition 8.2 (anisotropic shape operator, cf. [22], [9]) . (i) The linear map S γp : T p M → T p M given by the n × n matrix S γ := − d ˜ ξ γ is called the anisotropic shape operatior of X .(ii) If S γp is proportional to the identity map, we say that p is anisotropic-umbilic. Definition 8.3 (anisotropic curvatures, cf. [22], [9]) . (i) The eigenvalues of S γ are calledthe anisotropic principal curvatures of X . We denote them by k γ , · · · , k γn .(ii) Let σ γr be the elementary symmetric functions of k γ , · · · , k γn , that is σ γr := X ≤ l < ··· 1. Hence, each r -th anisotropicmean curvature of ξ γ is ( − r . Lemma 8.1. (i) ([8]) If A := dξ = D γ + γ · is positive definite, then all of the anisotropicprincipal curvatures of X : M → R n +1 are real.(ii) k γi is not a real value in general. However, each H γr is always a real valued functionon M .Proof. The outline of the proof of (i) was given in [8, p.699]. We will give a detailed proofin § B. (ii) is proved as follows. Denote by I n the n × n identity matrix. We computedet( I n + τ S γ ) = (1 + τ k γ ) · · · (1 + τ k γn ) (37)= n X r =0 τ r X ≤ l < ··· l r ≤ n k γl · · · k γl r (38)= n X r =0 τ r σ γr = n X r =0 ( n C r ) H γr τ r , (39)which shows that each H γr is a real valued function on M . ✷ Remark 8.2 (cf. [15]) . Take any point p ∈ M o . Assume that ν ( p ) is a regular point of( D γ + γ · p ) of X at p . Let { e , · · · , e n } be a locally defined frame on S n such that ( D γ + γ · e i = (1 /µ i ) e i , where µ i are theprincipal curvatures of ξ γ with respect to ν . Note that the basis { e , · · · , e n } at ν ( p ) alsoserves as an orthogonal basis for the tangent hyperplane of X at p . Let ( − w ij ) be thematrix representing dν with respect to this basis. Then − S γ = dξ γ ◦ dν = ( D γ + γ · dν = − w /µ · · · − w n /µ · · · · ·· · · · ·· · · · ·− w n /µ n · · · − w nn /µ n . This with (69) gives Λ = (1 /n )( w /µ + · · · + w nn /µ n ) . (40)For later use, we give some notations and formulas using local coordinates ( u , · · · , u n )in M . Set as usual g ij := h X i , X j i , h ij := −h ν i , X j i = −h X i , ν j i , where X i = X u i , etc . At any regular point of X , we set ( g ij ) = ( g ij ) − . 21y using the Einstein summation convention, we have ν i = − h ij g jk X k , and dν can be represented as an n × n matrix dν = − ( h ij )( g ij ) . For the Cahn-Hoffman map ξ : S n → R n +1 , consider its differential( dξ ) ν : T ν S n → T ξ ( ν ) R n +1 = R n +1 , dξ = D γ + γ · . For the Cahn-Hoffman field ˜ ξ := ξ ◦ ν : M → R n +1 along X , set − ˜ h ij := h ˜ ξ i , X j i , ˜ ξ i := ˜ ξ u i . (41)Then, it is easy to show that ˜ ξ i = − ˜ h il g lj X j (42)holds. In fact, since ˜ ξ i is perpendicular to ν (Proposition 5.1), we can write˜ ξ i = a ji X j , (43)which gives − ˜ h il = h ˜ ξ i , X l i = h a ji X j , X l i = a ji g jl , and so we have ( − ˜ h il ) = ( a ji )( g jl ) . Hence ( a ji ) = ( − ˜ h il )( g lj ) , which gives (42). Therefore d ˜ ξ is represented as an n × n matrix d ˜ ξ = − (˜ h ij )( g ij ) . (44)Since S γ = − d ˜ ξ , we have the following. Lemma 8.2. S γ = (˜ h ij )( g ij ) holds. In this section we assume that γ : S n → R > is of C .Let M be an oriented compact connected n -dimensional C ∞ manifold with smoothboundary ∂M . We assume that a map X : M → R n +1 satisfies (A1), (A2), and (A3) in § r = 2, X i = X , M i = M , and ν i = ν .22ecall that the Cahn-Hoffman field along X (for γ ) is defined as˜ ξ := ˜ ξ γ := ξ γ ◦ ν : M → R n +1 , (45)where ξ γ ( ν ) = Dγ + γ ( ν ) ν , ( ν ∈ S n ), is the Cahn-Hoffman map for γ . Remark 9.1. The anisotropic mean curvature Λ of X at any regular point of X can berepresented as Λ = 1 n ( − div M Dγ + nHγ ) , where H is the mean curvature of X (cf. [22], [15]).First we give the first variation formula of the anisotropic surface energy for immersions. Lemma 9.1. Assume that X : M → R n +1 is an immersion and let X ǫ : M → R n +1 be avariation of X = X . We assume for simplicity that X ǫ is of C ∞ in ǫ . Set δX := ∂X ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 , ψ := (cid:10) δX, ν (cid:11) . Then the first variation of the anisotropic energy F γ is given as follows. δ F γ := d F γ ( X ǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = − Z M n Λ ψ dA − I ∂M h ˜ ξ, −h δX, ν i N + h δX, N i ν i d ˜ s, (46) where dA is the n -dimensional volume form of M induced by X , N is the outward-pointingunit conormal along ∂M , and d ˜ s is the ( n − -dimensional volume form of ∂M inducedby X . Denote by R the π/ -rotation on the ( N, ν ) -plane. Denote by p the projection from R n +1 to the ( N, ν ) -plane. Then δ F γ = − Z M n Λ ψ dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s. (47) Proof. We can write X ǫ = X + ǫ ( η + ψν ) + O ( ǫ ) , where η is the tangential component and ψν is the normal component of the variationvector field δX of X ǫ . Then the first variation of the anisotropic energy F γ is computed asfollows. δ F γ := d F γ ( X ǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = Z M h Dγ, −∇ ψ + dν ( η ) i + γ ( − nHψ + div η ) dA (48)= Z M h Dγ, −∇ ψ i − nHγψ dA + Z M h Dγ, dν ( η ) i + γ div η dA (49)= Z M ψ (div M Dγ − nHγ ) dA + I ∂M − ψ h Dγ, N i + γ h η, N i d ˜ s. (50)23e compute the integrand of the boundary integral in (50): f := − ψ h Dγ, N i + γ h η, N i (51)= −h δX, ν ih ˜ ξ, N i + h ˜ ξ, ν ih δX, N i (52)= h ˜ ξ, −h δX, ν i N + h δX, N i ν i . (53)(53) with (50) gives (46).From (52), we obtain also the following: f = h δX, −h ˜ ξ, N i ν + h ˜ ξ, ν i N i . (54)By using R ( N ) = ν, R ( ν ) = − N, (55)we compute f = h δX, −h ˜ ξ, N i ν + h ˜ ξ, ν i N i (56)= h δX, −h ˜ ξ, N i R ( N ) − h ˜ ξ, ν i R ( ν ) i (57)= h δX, − R ( h ˜ ξ, N i N + h ˜ ξ, ν i ν ) i (58)= h δX, − R ( p ( ˜ ξ )) i . (59)(50) with (59) gives δ F γ = Z M ψ (div M Dγ − nHγ ) dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s (60)= − Z M n Λ ψ dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s, (61)which gives (47). ✷ Next we give the first variation formula of the anisotropic surface energy for piecewisesmooth weak immersions, which is a generalization of Lemma 9.1. Proposition 9.1. We assume that the map X : M → R n +1 satisfies (A1), (A2), and (A3)in § r = 2 , X i = X , M i = M , and ν i = ν . Let X ǫ : M → R n +1 , ( ǫ ∈ J := [ − ǫ , ǫ ] ),be a variation of X , that is, ǫ > and X = X . We assume for simplicity that X ǫ is of C ∞ in ǫ . Then, each X ǫ satisfies (A2) and (A3) in § r = 2 , X i = X ǫ , M i = M , and ν i = ν ǫ . We also assume that, for each ǫ ∈ J , the anisotropic mean curvature Λ ǫ of X ǫ isbounded on M o . Set δX := ∂X ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 , ψ := (cid:10) δX, ν (cid:11) . Then the first variation of the anisotropic energy F γ is given as follows. δ F γ = d F γ ( X ǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = − Z M n Λ ψ dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s, (62) where dA , d ˜ s , R , p are the same as those in Lemma 9.1. § n + 1)-dimensional volume enclosedby X ǫ is given as follows (cf. [4]). δV = Z M ψ dA. (63)Now let X : M = ∪ ki =1 M i → R n +1 be a piecewise- C weak immersion (for definition,see § 2) with unit normal ν i : M i → S n . We will use the following notations. X i := X | M i , ˜ ξ i := ξ ◦ ν i : M i → R n +1 . Denote by N i the outward-pointing unit conormal along ∂M i . Denote by R i the π/ N i , ν i )-plane. Denote by p i the projection from R n +1 to the ( N i , ν i )-plane.(47) together with (62), (63) gives the Euler-Lagrange equations in the following Propo-sition 9.2. Proposition 9.2 (Euler-Lagrange equations. For n = 2, see B. Palmer [21]) . A piecewise- C weak immersion X : M = ∪ ki =1 M i → R n +1 is a critical point of the anisotropic energy F γ ( X ) = Z M γ ( ν ) dA for ( n + 1) -dimensional volume-preserving variations if and only if(i) The anisotropic mean curvature Λ of X is constant on M \ S ( X ) , and(ii) ˜ ξ i ( ζ ) − ˜ ξ j ( ζ ) ∈ T ζ M i ∩ T ζ M j = T ζ ( ∂M i ∩ ∂M j ) holds at any ζ ∈ ∂M i ∩ ∂M j , herea tangent space of a submanifold of R n +1 is naturally identified with a linear subspace of R n +1 .Proof. The first term of the right-hand side of (47) together with (63) clearly gives (i). Bythe second term of the right-hand side of (62), the condition on ∂M i ∩ ∂M j is R i ( p i ( ˜ ξ i )) + R j ( p j ( ˜ ξ j )) = 0 . (64)Since, at any point ζ ∈ ∂M i ∩ ∂M j , ( N i , ν i )-plane coincides with ( N j , ν j )-plane, (64) isequivalent to p i ( ˜ ξ i ) − p j ( ˜ ξ j ) = 0 . (65)Again since, at any point ζ ∈ ∂M i ∩ ∂M j , ( N i , ν i )-plane coincides with ( N j , ν j )-plane, (65)implies (ii). ✷ Corollary 9.1 (Euler-Lagrange equations for the isotropic case) . A piecewise- C weakimmersion X : M = ∪ ki =1 M i → R n +1 is a critical point of the volume (that is the n -dimensional area) A ( X ) = Z M dA for ( n + 1) -dimensional volume-preserving variations ifand only if(i) The mean curvature H of X is constant on M , and(ii) X : M → R n +1 is a C ω immersion. roof. (i) is an immediate consequence of Proposition 9.2 (i).We prove (ii). Proposition 9.2 (ii) implies that ν i ( ζ ) − ν j ( ζ ) ∈ T ζ M i ∩ T ζ M j = T ζ ( ∂M i ∩ ∂M j ) (66)holds at any ζ ∈ ∂M i ∩ ∂M j . However, both of ν i ( ζ ) and ν j ( ζ ) are orthogonal to T ζ ( ∂M i ∩ ∂M j ), which implies that ν i ( ζ ) − ν j ( ζ ) = 0. Therefore, ν is continuous on M . Because ofthe ellipticity of the equation H = constant, X : M → R n +1 is a C ω immersion. ✷ Definition 9.1. A piecewise- C weak immersion X : M = ∪ ki =1 M i → R n +1 is called ahypersurface with constant anisotropic mean curvature (CAMC for short) if X satisfies theconditions (i) and (ii) in Proposition 9.2.If γ is convex, then we have the following striking result that the Cahn-Hoffman fieldalong any piecewise- C CAMC hypersurface is defined continuously over the whole hyper-surface as follows. Theorem 9.1. Assume γ : S n → R > is of C and the convex integrand of its Wulff shape W γ . Let X : M = ∪ ki =1 M i → R n +1 be a closed piecewise- C CAMC hypersurface withunit normal ν i : M i → S n , ( i = 1 , · · · k ). Then, the collection of the Cahn-Hoffman fields ˜ ξ i := ξ ◦ ν i : M i → R n +1 , ( i = 1 , · · · k ) defines a C map ˜ ξ : M → R n +1 .Proof . Since γ is convex and of C , W γ = ˆ W γ and it is strictly convex ([7]), that is, forany two different points q , q ∈ W γ , the straight line containing q , q intersects W γ onlyat q , q .We consider the boundary condition in Proposition 9.2 (ii), which is˜ ξ i ( q ) − ˜ ξ j ( q ) ∈ T q ( ∂M i ∩ ∂M j ) , ∀ q ∈ ∂M i ∩ ∂M j . (67)Take an arbitrary point q ∈ ∂ Σ i ∩ ∂ Σ j . Assume that ˜ ξ i ( q ) = ˜ ξ j ( q ). Then, from (67), wehave h ˜ ξ i ( q ) − ˜ ξ j ( q ) , ν i ( q ) i = 0 = h ˜ ξ i ( q ) − ˜ ξ j ( q ) , ν j ( q ) i . Hence, ( ˜ ξ i ( q ) − ˜ ξ j ( q )) ∈ T ˜ ξ i ( q ) W, ( ˜ ξ i ( q ) − ˜ ξ j ( q )) ∈ T ˜ ξ j ( q ) W (68)holds. However, (68) contradicts the fact that W is strictly convex. Hence ˜ ξ i ( q ) = ˜ ξ j ( q )holds, and we have proved the desired result. ✷ Remark 9.2. It is shown that, at any regular point of X , it holds thatΛ = − n trace M ( D γ + γ · ◦ dν = − n trace M d ( ξ γ ◦ ν ) = − n trace M d ( ˜ ξ γ ) (69)(cf. [15]). Hence, at points where γ is strictly convex, by (69), the equation “Λ = constant”is elliptic. 26 emark 9.3. Here we give a known representation formula for the anisotropic mean cur-vature for a graph for readers’ convenience. We denoted the homogeneous extension of γ : S n → R > by γ : R n +1 → R ≥ . Let X be a graph X : D → R , X ( u , u ) = ( u , u , f ( u , u ))of a C ∞ function f : D ( ⊂ R ) → R . Set f i := f u i , f ij := f u i u j , Df := ( f , f ) . Then, Λ = 12 X i,j =1 , γ x i x j (cid:12)(cid:12)(cid:12) ( − Df, f u i u j holds. In the special case where γ ( Y ) ≡ | Y | , that is, γ ≡ f ) ) f − f f f + (1 + ( f ) ) f f ) + ( f ) + 1) / holds. Hence, the equation “Λ ≡ constant” is elliptic or hyperbolic depends on the 2 × γ x i x j (cid:12)(cid:12)(cid:12) ( − Df, ) i,j =1 , . Proposition 9.3. The anisotropic mean curvature of the Cahn-Hoffman map ξ : S n → R n +1 is − at any regular point ν ∈ S n with respect to the unit normal ν . Hence, particularlythe anisotropic mean curvature of the Wulff shape (for the outward-pointing unit normal)is − at any regular point.Proof . Since ξ − gives the unit normal vector field ν ξ for the Cahn-Hoffman map ξ : S n → R n +1 (Corollary 5.1), using (69), we haveΛ = − n trace( d ( ξ ◦ ξ − )) = − n trace( I n ) = − , (70)where I n is the n × n identity matrix. ✷ Proposition 9.4. Set A := D γ + γ · . Take the unit normal vector ˜ ν of the Cahn-Hoffman map ξ = ξ γ so that ˜ ν = ν holds at any point ν ∈ S n where det A > holds.Let { e , · · · , e n } be an orthonormal basis of T ν S n such that { ˜ ν, f , · · · , f n } is compatiblewith the canonical orientation of R n +1 , where f j := ( d ν ξ )( e j ) . Then, at each point ν ∈ S n satisfying det A < , ˜ ν = − ν holds. If Λ is the anisotropic mean curvature of ξ with respectto ˜ ν , then Λ = ( − , (det A > , , (det A < . (71)Proposition 9.4 gives Corollary 9.2. ξ γ is not CAMC in general. emark 9.4. Example 7.7 gives a good example for Corollary 9.2. The closed curve withself-intersection in Figure 6 is the image ξ γ ( S ) of ξ γ . At each regular point on each smootharc containing a solid arc, Λ = − 1, while at other regular points Λ = 1. Hence, ξ γ ( S ) isnot CAMC. Proposition 9.5. There is no closed piecewise-smooth weakly immersed hypersurface in R n +1 that is a critical point of F γ . This means that there is no closed piecewise-smoothweakly immersed hypersurface in R n +1 whose anisotropic mean curvature is constant zero.Proof. Let X : M n → R n +1 be a piecewise-smooth weakly immersed hypersurface in R n +1 .Consider the homotheties X ǫ := (1 + ǫ ) X of X . Then, since F γ ( X ǫ ) = (1 + ǫ ) n F γ ( X ) > ddǫ (cid:12)(cid:12)(cid:12) ǫ =0 F γ ( X ǫ ) = n F γ ( X ) > X is not a critical point of F γ ( X ). 10 Anisotropic parallel hypersurface and Steiner-type for-mula We assume that γ : S n → R > is of C , and use the same notations as in § 8. ‘ Anisotropicparallel hypersurface ’ is a generalization of parallel hypersurface which is defined as follows. Definition 10.1 (Anisotropic parallel hypersurface, cf. [22]) . Let X be a piecewise- C weak immersion. For any real number t , we call the map X t := X + t ˜ ξ : M → R n +1 theanisotropic parallel deformation of X of height t . If X t is a piecewise- C weak immersion,then we call it the anisotropic parallel hypersurface of X of height t .The anisotropic energy F γ ( X t ) of the anisotropic parallel hypersurface X t := X + t ˜ ξ isa polynomial of t of degree at most n as follows. Theorem 10.1 (Steiner-type formula) . Assume that γ : S n → R > is of C . Assume that X : M = ∪ ki =1 M i → R n +1 is a piecewise- C weak immersion. Consider anisotropic parallelhypersurfaces X t = X + t ˜ ξ : M \ S ( X ) → R n +1 , where S ( X ) is the set of singular points of X . Then the following integral formula holds. F γ ( X t ) = Z M γ ( ν ) dA t = Z M γ ( ν )(1 − tk γ ) · · · (1 − tk γn ) dA, (72) where R M means P ki =1 R M i . Moreover, if the r -th anisotropic mean curvature of X for γ isintegrable on M for r = 1 , · · · , n , then F γ ( X t ) = n X r =0 ( − r t r ( n C r ) Z M γ ( ν ) H γr dA (73) holds Corollary 10.1 (Steiner’s formula) . Assume that X : M → R is an immersion froma -dimensional oriented connected compact C ∞ manifold M into R with unit normal ν : M → S . Denote by H , K the mean and the Gaussian curvature of X , respectively.Then the area A ( X t ) of the parallel surface X t = X + tν of height t has the followingrepresentation. A ( X t ) = A ( X ) − t Z M H dA + t Z M K dA. (74)Now assume that X t := X + t ˜ ξ : M → R n +1 is an immersion. Let ( u , · · · , u n ) be localcoordinates in M . We compute the volume form dA t = (det( g tij )) / du · · · du n , (75)where g tij := h ( X t ) i , ( X t ) j i , ( X t ) i := ( X t ) u i . We compute g tij = h X ti , X tj i = h X i + t ˜ ξ i , X j + t ˜ ξ j i = h X i , X j i + t ( h X i , ˜ ξ j i + h ˜ ξ i , X j i ) + t h ˜ ξ i , ˜ ξ j i . (76)By using (42), we have h ˜ ξ i , ˜ ξ j i = h− ˜ h il g lk X k , − ˜ h ja g ab X b i = ˜ h il g lk ˜ h ja g ab g kb = ˜ h il g lk ˜ h jk . (77)(76) with (41) and (77) gives g tij = g ij − t (˜ h ij + ˜ h ji ) + t ˜ h il g lk ˜ h jk . (78)Define three n × n matrices as follows: G := ( g ij ) , ˜ B := (˜ h ij ) , G t := ( g tij ) . Note that ˜ B is not symmetric in general.From (78), we havedet G t = det G · det (cid:16) I n − t ( ˜ BG − + t ˜ BG − ) + t ( ˜ BG − t ˜ BG − ) , (cid:17) (79)= det G · det( I n − t ˜ BG − ) · det( I n − t t ˜ BG − ) , (80)here t ˜ B is the transposed matrix of ˜ B . Lemma 10.1. (i) det( I n − t ˜ BG − ) = det( I n − t t ˜ BG − ) . (81) Hence, det( I n − t ˜ BG − ) and det( I n − t t ˜ BG − ) have the same eigenvalues. ii) det( I n − t ˜ BG − ) = det( I n − t t ˜ BG − ) (82)= (1 − tk γ ) · · · (1 − tk γn ) (83)= n X r =0 ( − r t r X ≤ l < ··· l r ≤ n k γl · · · k γl r (84)= n X r =0 ( − r t r σ γr = n X r =0 ( − r t r ( n C r ) H γr . (85) (iii) dA t = (1 − tk γ ) · · · (1 − tk γn ) dA = n X r =0 ( − r t r ( n C r ) H γr dA. Proof. First we prove (i). Observedet( I n − t ˜ BG − ) = det t ( I n − t ˜ BG − ) = det( t I n − t t G − t ˜ B ) (86)= det( I n − tG − t ˜ B ) = det G · det( I n − tG − t ˜ B ) · det G − (87)= det( I n − t t ˜ BG − ) , (88)which gives (i).(ii) is given by (39) by replacing τ with − t because S γ = (˜ h ij )( g ij ) = ˜ BG − (Lemma8.2).(iii) is given by (75), (80) and (ii) of this lemma. ✷ 11 Minkowski-type formula In this section we give another integral formula that is a generalization of the Minkowskiformula (see [8] for smooth case). Theorem 11.1 (Minkowski-type formula) . Assume that γ : S n → R > is of C . As-sume that X : M = ∪ ki =1 M i → R n +1 is a closed piecewise- C weak immersion whose r -thanisotropic mean curvature for γ is integrable on M for r = 1 , · · · , n . Assume also that X satisfies the following condition. ˜ ξ i ( ζ ) = ˜ ξ j ( ζ ) , ∀ ζ ∈ ∂M i ∩ ∂M j , here ˜ ξ i := ξ ◦ ν i : M i → R n +1 is the Cahn-Hoffman field along X | M i , and the tangent spaceof a submanifold of R n +1 is naturally identified with a linear subspace of R n +1 .Then the following integral formula holds. Z M ( γ ( ν ) H γr + h X, ν i H γr +1 ) dA = 0 , r = 0 , · · · , n − , (89)30 here H γr is the r -th anisotropic mean curvature of X (Definition 8.3). In the special casewhere r = 0 , we have Z M ( γ ( ν ) + Λ h X, ν i ) dA = 0 . (90)Before proving Theorem 11.1, we give a useful lemma: Lemma 11.1. Assume the same assumption as Theorem 11.1. For t ∈ R , | t | << , set X t := X + t ˜ ξ. Then, ddt V ( X t ) = Z M γ ( ν ) dA t = F γ ( X t ) (91)= n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA (92) holds.Proof. The unit normal along X t is the same as that of X . Hence, using (63) and Theorem10.1, we obtain ddt V ( X t ) = Z M h ˜ ξ, ν i dA t = Z M h Dγ | ν + γ ( ν ) ν, ν i dA t = Z M γ ( ν ) dA t = F γ ( X t )= n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA. ✷ Proof of Theorem 11.1. For t ∈ R , | t | << 1, set X t := X + t ˜ ξ. Since the unit normal along X t is the same as that of X , it holds that V ( X t ) = 1 n + 1 Z M h X t , ν i dA t = 1 n + 1 Z M h X + t ˜ ξ, ν i dA t . (93)Hence we have ddt V ( X t ) = 1 n + 1 Z M h ˜ ξ, ν i dA t + 1 n + 1 Z M h X + t ˜ ξ, ν i ∂ ( dA t ) ∂t (94)=: I + II. (95)31e compute, using Lemma 10.1 (iii), to obtain II := 1 n + 1 Z M h X + t ˜ ξ, ν i ∂ ( dA t ) ∂t = 1 n + 1 Z M ( h X, ν i + tγ ( ν )) ∂∂t n X r =0 ( − r t r ( n C r ) H γr dA = 1 n + 1 Z M ( h X, ν i + tγ ( ν )) n X r =1 ( − r rt r − ( n C r ) H γr dA = 1 n + 1 n X r =1 ( − r rt r − ( n C r ) Z M h X, ν i H γr dA + 1 n + 1 n X r =1 ( − r rt r ( n C r ) Z M γ ( ν ) H γr dA = 1 n + 1 n − X r =0 ( − r +1 ( r + 1) t r ( n C r +1 ) Z M h X, ν i H γr +1 dA + 1 n + 1 n X r =1 ( − r rt r ( n C r ) Z M γ ( ν ) H γr dA. (96)Again, by using Lemma 10.1 (iii), we obtain I := 1 n + 1 Z M h ˜ ξ, ν i dA t = 1 n + 1 Z M h Dγ + γν, ν i dA t = 1 n + 1 Z M γ dA t = 1 n + 1 n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA. (97)On the other hand, by Lemma 11.1, we have ddt V ( X t ) = n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA. (98)The equalities (95), (96), (97) and (98) give n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA = 1 n + 1 n X r =0 ( − r ( n C r ) t r Z M γ ( ν ) H γr dA + 1 n + 1 n − X r =0 ( − r +1 ( r + 1) t r ( n C r +1 ) Z M h X, ν i H γr +1 dA + 1 n + 1 n X r =1 ( − r rt r ( n C r ) Z M γ ( ν ) H γr dA, (99)that is, n X r =0 ( − r ( n − r )( n C r ) t r Z M γ ( ν ) H γr dA = − n − X r =0 ( − r ( r + 1) t r ( n C r +1 ) Z M h X, ν i H γr +1 dA, (100)which gives n − X r =0 ( − r ( n − r )( n C r ) t r Z M ( γ ( ν ) H γr + h X, ν i H γr +1 ) dA = 0 . (101)32101) implies Z M ( γ ( ν ) H γr + h X, ν i H γr +1 ) dA = 0 , r = 0 , · · · , n − , (102)which is the desired result. ✷ Remark 11.1. If we want to have only the formula (90), the following proof works.For t ∈ R , | t | << , set X t := (1 + t ) X. Then we have f ( t ) := F γ ( X t ) = Z M γ ( ν )(1 + t ) n dA, and f ′ ( t ) = n (1 + t ) n − Z M γ ( ν ) dA. (103) Hence f ′ (0) = n Z M γ ( ν ) dA (104) holds. On the other hand, the first variation formula (Lemma 9.1) gives f ′ (0) = − n Z M Λ D ∂X t ∂t (cid:12)(cid:12)(cid:12) t =0 , ν E dA = − n Z M Λ h X, ν i dA. (105) (104) with (105) gives (90). 12 Proof of Theorem 1.1 First we give an outline of the proof of Theorem 1.1. The idea is to generalize the proof ofthe uniqueness of stable closed CMC hypersurface in R n +1 given by [26], which was usedalso in [20] and [21]. In order to prove Theorem 1.1, we first recall that the Cahn-Hoffmanfield ˜ ξ along a closed CAMC hypersurface X : M = ∪ ki =1 M i → R n +1 is well-defined onthe whole of M (Theorem 9.1). Then, we consider the anisotropic parallel hypersurfaces X t := X + t ˜ ξ , ( t ∈ R , | t | << 1) of X . By taking homotheties of X t if necessary, we havea volume-preserving variation Y t = µ ( t ) X t = µ ( t )( X + t ˜ ξ ), ( µ ( t ) > µ (0) = 1) of X . Byusing the Steiner-type formula (Theorem 10.1) and the Minkowski-type formula (Theorem11.1), we prove that d F γ ( Y t ) dt (cid:12)(cid:12)(cid:12) t =0 = − n Z M γ ( ν ) X ≤ i Assume that X : M = ∪ ki =1 M i → R n +1 is a closed piecewise- C CAMC hypersurface for γ : S n → R > . Because of Lemma 9.1, the Cahn-Hoffman field˜ ξ : M → R n +1 along X can be defined on M . Consider the anisotropic parallel hypersurfaces X t : M → R n +1 of X , that is X t := X + t ˜ ξ, t ∈ R , | t | << . By taking homotheties of X t if necessary, we have a volume-preserving variation Y t of X which is represented as follows. Y t = µ ( t ) X t = µ ( t )( X + t ˜ ξ ) , µ ( t ) > , µ (0) = 1 . Set F := F γ ( X ) , V := V ( X ) . And set f ( t ) := F γ ( Y t ) , v ( t ) := V ( Y t ) . Then, f ( t ) = ( µ ( t )) n F γ ( X t ) , v ( t ) = ( µ ( t )) n +1 V ( X t ) , f (0) = F , v (0) = V . (106)We will compute f ′′ (0). In order to do it, we will compute µ ′ (0) and µ ′′ (0) by using v ( t ) ≡ V .From (106), we have v ′ ( t ) = ( n + 1)( µ ( t )) n µ ′ ( t ) V ( X t ) + ( µ ( t )) n +1 dV ( X t ) dt , (107)34 ′′ ( t ) = ( n + 1) n ( µ ( t )) n − ( µ ′ ( t )) V ( X t ) + ( n + 1)( µ ( t )) n µ ′′ ( t ) V ( X t )+2( n + 1)( µ ( t )) n µ ′ ( t ) dV ( X t ) dt + ( µ ( t )) n +1 d V ( X t ) dt . (108)From (91), dV ( X t ) dt (cid:12)(cid:12)(cid:12) t =0 = F . (109)Using v ′ (0) = 0, (107) with (109) gives µ ′ (0) = − F ( n + 1) V . (110)Using (92), we obtain d V ( X t ) dt (cid:12)(cid:12)(cid:12) t =0 = − ( n C ) Z M γ Λ dA = − n Λ F . (111)From (90) in Theorem 11.1, we have F + Λ( n + 1) V = 0 , and hence Λ = − F ( n + 1) V . (112)Using (108), (109), (110), (111) and (112), we obtain0 = v ′′ (0) = − n + 1 · F V + ( n + 1) V µ ′′ (0) , (113)which gives µ ′′ (0) = 2( n + 1) · F V . (114)From (106), we have f ′ ( t ) = n ( µ ( t )) n − µ ′ ( t ) F γ ( X t ) + ( µ ( t )) n d F γ ( X t ) dt , (115) f ′′ ( t ) = n ( n − µ ( t )) n − ( µ ′ ( t )) F γ ( X t ) + n ( µ ( t )) n − µ ′′ ( t ) F γ ( X t )+2 n ( µ ( t )) n − µ ′ ( t ) d F γ ( X t ) dt + ( µ ( t )) n d F γ ( X t ) dt . (116)Using (73) in Theorem 10.1, (110), (112), (114) and (116), we obtain f ′′ (0) = − n ( n − n + 1) · F V + 2( n C ) Z M γ ( ν ) H γ dA = − n ( n − F + 2 Z M γ ( ν ) X ≤ i 0. Because M is closed and W has anisotropic mean curvature − X ( M ) = (1 / | Λ | ) W. ✷ 13 Proof of Proposition 9.1 For simplicity, we write M instead of M . We take an exhaustion { M t } ≤ t< of M o . Thatis, (i) M t ⊂ M o , ∀ t ∈ [0 , K ⊂ M o , there exists a number t ∈ (0 , 1) such that M t ⊃ K for all t ∈ ( t , M := M .Let X ǫ : M → R n +1 , ( ǫ ∈ J := [ − ǫ , ǫ ], ǫ > X . Set X tǫ := X ǫ | M t .And set ψ ǫ := h ∂X ǫ ∂ǫ , ν ǫ i . Denote by Λ ǫ the anisotropic mean curvature of X ǫ , and by dA ǫ the volume form of M with the metric induced by X ǫ . Note that Λ ǫ is defined at regular points of X ǫ . Also notethat, if X ǫ has singular points in ∂M , then dA ǫ is singular but it is well-defined at thesepoints. From Lemma 9.1, we have δ F γ t := d F γ ( X tǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = − Z M t n Λ ψ dA − I ∂M t h δX, R ( p ( ˜ ξ )) i d ˜ s =: I ( t ) + II ( t ) (121)for t ∈ [0 , ǫ ∈ [ − ǫ , ∪ (0 , ǫ ] and t ∈ [0 , f ( ǫ, t ) := Z M t ǫ (cid:16) γ ( ν ǫ ) dA ǫ − γ ( ν ) dA (cid:17) . (122)Then, (121) implies thatlim ǫ → f ( ǫ, t ) = d F γ ( X tǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = I ( t ) + II ( t ) , t ∈ [0 , 1) (123)holds. 36e also have the following. For any ǫ ∈ ( − ǫ , ∪ (0 , ǫ ) and t ∈ (0 , θ ∈ (0 , 1) such that f ( ǫ, t ) = d F γ ( X tσ ) dσ (cid:12)(cid:12)(cid:12) σ = θǫ = − Z M t n Λ θǫ ψ θǫ dA θǫ − I ∂M t h ∂X σ ∂σ (cid:12)(cid:12)(cid:12) σ = θǫ , R ( p ( ˜ ξ θǫ )) i d ˜ s θǫ =: I θǫ ( t )+ II θǫ ( t )(124)holds. Set c ( ǫ ) = sup M | Λ θǫ ψ θǫ − Λ ψ | , (125) c = sup M | Λ ψ | . (126)Because Λ ǫ is bounded on M for any ǫ , c ( ǫ ) and c are finite numbers.In order to prove Proposition 9.1, we prepare several claims. Claim 13.1. The convergence in (123) is uniform with respect to t .Proof. We compute1 n | I θǫ ( t ) − I ( t ) | = (cid:12)(cid:12)(cid:12)Z M t (Λ θǫ ψ θǫ dA θǫ − Λ ψ dA ) (cid:12)(cid:12)(cid:12) (127) ≤ Z M t | Λ θǫ ψ θǫ dA θǫ − Λ ψ dA | (128) ≤ Z M t | Λ θǫ ψ θǫ − Λ ψ | dA θǫ + Z M t | Λ ψ | · | dA θǫ − dA | (129) ≤ c ( ǫ ) Z M t dA θǫ + c Z M t | dA θǫ − dA | (130) ≤ c ( ǫ ) Z M dA θǫ + c Z M | dA θǫ − dA | . (131)Since Z M dA θǫ is bounded andlim ǫ → c ( ǫ ) = 0 , lim ǫ → Z M | dA θǫ − dA | = 0hold, | I θǫ ( t ) − I ( t ) | → , as ǫ → t . Next we have1 n | II θǫ ( t ) − II ( t ) | = (cid:12)(cid:12)(cid:12)I ∂M t h δX | θǫ , R ( p ( ˜ ξ θǫ )) i d ˜ s θǫ − I ∂M t h δX | , R ( p ( ˜ ξ )) i d ˜ s (cid:12)(cid:12)(cid:12) (132) ≤ I ∂M t (cid:12)(cid:12)(cid:12) h δX | θǫ , R ( p ( ˜ ξ θǫ )) i d ˜ s θǫ − h δX | , R ( p ( ˜ ξ )) i d ˜ s (cid:12)(cid:12)(cid:12) (133) ≤ I ∂M t h(cid:12)(cid:12)(cid:12) h δX | θǫ , R ( p ( ˜ ξ θǫ )) i − h δX | , R ( p ( ˜ ξ )) i (cid:12)(cid:12)(cid:12) d ˜ s θǫ (134)+ |h δX | , R ( p ( ˜ ξ )) i| · | d ˜ s θǫ − d ˜ s | i (135) ≤ c ( ǫ ) I ∂M t d ˜ s θǫ + c I ∂M t | d ˜ s θǫ − d ˜ s | , (136)37here c ( ǫ ) := max ≤ t ≤ max ∂M t (cid:12)(cid:12)(cid:12) h δX | θǫ , R ( p ( ˜ ξ θǫ )) i − h δX | , R ( p ( ˜ ξ )) i (cid:12)(cid:12)(cid:12) , (137) c = max ≤ t ≤ max ∂M t (cid:12)(cid:12)(cid:12) h δX | , R ( p ( ˜ ξ )) i (cid:12)(cid:12)(cid:12) . (138)Since nI ∂M t d ˜ s θǫ ; t ∈ [0 , , ǫ ∈ [ − ǫ , ǫ ] o is bounded, and sincelim ǫ → c ( ǫ ) = 0 , lim ǫ → max ≤ t ≤ I ∂M t | d ˜ s θǫ − d ˜ s | = 0hold, | II θǫ ( t ) − II ( t ) | → , as ǫ → t . Therefore, the convergence in (123) is uniform with respect to t . ✷ Claim 13.2. The improper integral Z M | Λ | dA converges.Proof. By assumption, the anisotropic mean curvature Λ of X is bounded on M . Hence,for any exhaustion { ˆ M s } ≤ s< of M (for the definition of exhaustion, see the beginning of § Z ˆ M s | Λ | dA is bounded and an increasing function of s . Therefore, lim s → − Z ˆ M s | Λ | dA converges, which implies the desired result. ✷ Claim 13.3. lim t → − I ( t ) and lim t → − II ( t ) converge. Hence lim t → − (cid:16) lim ǫ → f ( ǫ, t ) (cid:17) converges. And we may write lim t → − (cid:16) lim ǫ → f ( ǫ, t ) (cid:17) = − Z M n Λ ψ dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s. (139) Proof. Recall I ( t ) = − Z M t n Λ ψ dA, II ( t ) = − I ∂M t h δX, R ( p ( ˜ ξ )) i d ˜ s. For 0 < t < t < 1, we have1 n | I ( t ) − I ( t ) | = (cid:12)(cid:12)(cid:12)Z M t \ M t Λ ψ dA (cid:12)(cid:12)(cid:12) ≤ Z M t \ M t (cid:12)(cid:12) Λ ψ (cid:12)(cid:12) dA ≤ max M | ψ | · Z M t \ M t (cid:12)(cid:12) Λ (cid:12)(cid:12) dA. (140)Since Z M | Λ | dA converges (Claim 13.2), (140) implies that, for any σ > 0, there exists anumber T ∈ (0 , 1) such that if T < t < t < 1, then | I ( t ) − I ( t ) | < σ . Hence lim t → − I ( t )converges. Convergence of lim t → − II ( t ) is obvious. ✷ The following claim is obviously true. 38 laim 13.4. For any ǫ ∈ ( − ǫ , ∪ (0 , ǫ ) , lim t → − f ( ǫ, t ) = Z M ǫ (cid:16) γ ( ν ǫ ) dA ǫ − γ ( ν ) dA (cid:17) (141) holds.Proof of Proposition 9.1. From Claims 13.1, 13.3, and 13.4, the limitlim ǫ → t → − f ( ǫ, t )exists, and lim ǫ → (cid:16) lim t → − f ( ǫ, t ) (cid:17) = lim ǫ → t → − f ( ǫ, t ) = lim t → − (cid:16) lim ǫ → f ( ǫ, t ) (cid:17) (142)holds. (142) together with (141), (139) implies d F γ ( X ǫ ) dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = − Z M n Λ ψ dA − I ∂M h δX, R ( p ( ˜ ξ )) i d ˜ s, (143)which completes the proof of Proposition 9.1. ✷ A Computations for Example 7.5 Set γ m (cos θ, sin θ ) = (cos m θ + sin m θ ) / (2 m ) (144)= γ m ( θ ) . (145)The homogeneous extension of γ is γ m ( x , x ) = ( x m + x m ) / (2 m ) . (146)And so, Dγ m ( x , x ) := (( γ m ) x , ( γ m ) x ) = ( x m + x m ) (1 / (2 m )) − ( x m − , x m − ) . (147)Hence, the Cahn-Hoffman map ξ m for γ m is computed as follows. ξ m (cos θ, sin θ ) := ( f m ( θ ) , g m ( θ )) (148):= D S γ m (cid:12)(cid:12)(cid:12) ν + γ m ( ν ) ν = Dγ m ( x , x ) | (cos θ, sin θ ) (149)= (cos m θ + sin m θ ) (1 / (2 m )) − (cos m − θ, sin m − θ ) . (150)Also, we have∆ γ m ( x , x ) := ( γ m ) x x + ( γ m ) x x = (2 m − x m − x m − ( x + x )( x m + x m ) (1 / (2 m )) − , (151) A m := dξ m := D γ m + γ m · γ m ( x , x ) (cid:12)(cid:12)(cid:12) (cos θ, sin θ ) = (2 m − 1) cos m − θ sin m − θ (cos m θ + sin m θ ) (1 / (2 m )) − . (152)39ence,(i) If m ≥ A m = 0 at θ = (1 / ℓπ , ( ℓ ∈ Z ).(ii) A m is positive-definite on S \ { (cos θ, sin θ ) | θ = (1 / ℓπ, ( ℓ ∈ Z ) } .Using (150), we compute the curvature κ m of ξ m with respect to the outward-pointingnormal ν and obtain the followings. κ m = − f ′ m g ′′ m + f ′′ m g ′ m (( f ′ m ) + ( g ′ m ) ) / , (153)(( f ′ m ) + ( g ′ m ) ) / = (2 m − 1) cos m − θ sin m − θ (cos m θ + sin m θ ) m − , (154) − f ′ m g ′′ m + f ′′ m g ′ m = − (2 m − cos m − θ sin m − θ (cos m θ + sin m θ ) m − (155)Using (153), (154), and (155), we obtain κ m = − m − − m +2 θ sin − m +2 θ (cos m θ + sin m θ ) − m . (156)Hence, for any ℓ ∈ Z and m ≥ 2, lim θ → ℓ π κ m ( θ ) = −∞ (157)holds. On the other hand, using (154) and (156), we obtain κ m ds = − f ′ m g ′′ m + f ′′ m g ′ m (( f ′ m ) + ( g ′ m ) ) / (( f ′ m ) + ( g ′ m ) ) / dθ = − dθ. (158)Hence Z π κ m ds = − Z π dθ = − π. (159)holds. However, (159) is trivial because ξ : S → R is a front with ν ∈ S . B Proof of Lemma 8.1 (i) By the same way as that outlined in [8, p.699], we can prove the desired result as follows.Let X : M n → R n +1 be an immersion with unit normal ν : M → S n . Let ( u , · · · , u n ) belocal coordinates in M . We use the same notations as in § 8. Note S γ = − d ˜ ξ = − dξ ◦ dν = A ( h ij )( g ij ) , A := D γ + γ · . By choosing a suitable coordinate system of M , Q := ( h ij )( g ij ) can be a symmetric matrix. A can be represented as an n × n matrix. Since A is symmetric, it has n real eigenvalues,which we denote by λ , · · · , λ n . Since A is positive definite, all of λ , · · · , λ n are positive.Since A is symmetric, it has an eigendecomposition P D t P , where P is a real orthogonalmatrix whose rows comprise an orthonormal basis of eigenvectors of A , and D is a realdiagonal matrix whose main diagonal contains the corresponding eigenvalues. Hence, we40an write D = λ · · · · · · λ · ·· · ·· · λ n − · · · · · · λ n = ( λ i δ ij ) i,j =1 , ··· ,n . Set ˜ D = √ λ · · · · · · √ λ · ·· · ·· · p λ n − · · · · · · √ λ n = ( p λ i δ ij ) i,j =1 , ··· ,n . Then, A = P D t P = P ˜ D ˜ D t P = P ˜ D t ˜ D t P = P ˜ D t ( P ˜ D ) . Set ˜ P := P ˜ D. Then, A = ˜ P t ˜ P . Note that ˜ P is a regular matrix. Hence, we havedet( tI − S γ ) = det( tI − AQ ) = det( tI − ˜ P t ˜ P Q ) (160)= det( ˜ P − ( tI − ˜ P t ˜ P Q ) ˜ P ) = det( tI − t ˜ P Q ˜ P ) . (161)Hence, S γ has the same eigenvalues as those of t ˜ P Q ˜ P . Since Q is symmetric, t ˜ P Q ˜ P is alsosymmetric. Therefore, S γ has n real eigenvalues. References [1] A. 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