Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature
aa r X i v : . [ m a t h . DG ] M a r Non-uniqueness of closed embedded non-smooth hypersurfaceswith constant anisotropic mean curvature
Yoshiki Jikumaru and Miyuki Koiso ∗ Abstract
An anisotropic surface energy is the integral of an energy density that depends onthe normal at each point over the considered surface, and it is a generalization of surfacearea. The minimizer of such an energy among all closed surfaces enclosing the samevolume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that,unlike the isotropic case, there exists an anisotropic energy density function such thatthere exist closed embedded equilibrium surfaces with genus zero in R each of whichis not (any homothety and translation of) the Wulff shape. We also give nontrivialself-similar shrinking solutions of anisotropic mean curvature flow. These results aregeneralized to hypersurfaces in R n +1 . An anisotropic surface energy is the integral of an energy density that depends on the surfacenormal, which was introduced by J. W. Gibbs (1839-1903) in order to model the shape ofsmall crystals ([17],[18]). The energy density function is given as a positive continuousfunction γ : S n → R > on the unit sphere S n = { ν ∈ R n +1 | k ν k = 1 } in the ( n + 1)-dimensional Euclidean space R n +1 . In this paper, for a technical reason, we study the casewhere γ is of C . When we study variational problems of such an energy, it is natural toconsider not only smooth surfaces but also surfaces with edges. And so let X : M → R n +1 be a piecewise- C weak immersion (the definition of piecewise- C weak immersion will begiven in § n -dimensional oriented connected compact C ∞ manifold M into R n +1 .The unit normal vector field ν along X is defined on M except a set with measure zero.The anisotropic energy F γ ( X ) of X is defined as F γ ( X ) = R M γ ( ν ) dA , where dA is the ∗ This work was partially supported by JSPS KAKENHI Grant Number JP18H04487. 2010 MathematicsSubject Classification:49Q10, 53C45, 53C42, 53C44. Key words and phrases: anisotropic mean curva-ture, anisotropic surface energy, Wulff shape, crystalline variational problem, Cahn-Hoffman vector field,anisotropic mean curvature flow. -dimensional volume form of M induced by X . In the special case where γ ≡ F γ ( X ) isthe usual n -dimensional volume of X that is the volume of M with metric induced by X .For any positive number V >
0, among all closed hypersurfaces in R n +1 enclosingthe same ( n + 1)-dimensional volume V , there exists a unique (up to translation in R n +1 )minimizer of F γ ([15]). Here a closed hypersurface means that the boundary (having tangentspace almost everywhere) of a set of positive Lebesgue measure. The minimizer for V =( n + 1) − R S n γ ( ν ) dS n is called the Wulff shape (for γ ) (see § W γ . W is the unit sphere S n . In general, W γ isconvex but not necessarily smooth.A piecewise- C weak immersion X : M → R n +1 is a critical point of F γ for all variationsthat preserve the enclosed ( n +1)-dimensional volume (we will call such a variation a volume-preserving variation) if and only if the anisotropic mean curvature of X is constant and X has a certain condition on its singular points (cf. § X is defined at each regular point of X as 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 (cf. [14], [10]).We call such X a CAMC (constant anisotropic mean curvature) hypersurface (Definition2.3). When γ ≡
1, Λ coincides with H and CAMC hypersurfaces are CMC (constant meancurvature) hypersurfaces (of C ω class).A natural question is ‘is any closed CAMC hypersurface X homothetic to the Wulffshape?’ The answer is not affirmative even in the case where γ ≡ X satisfies one of the following conditions (I)-(III), the image of X isa homothety of the Wulff shape.(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 .Here a CAMC hypersurface is said to be stable if the second variation of the energy F γ forany volume-preserving variation is nonnegative. If we assume that the Wulff shape W γ isa smooth strictly convex hypersurface, any closed CAMC hypersurface X is also smoothand the above expectation is correct, which was proved by [1], [3], [6] for γ ≡
1, and by [5],[12], [11] and [2] for general γ .If γ has less regularity or less convexity, the Wulff shape and CAMC hypersurfaces canhave “edges”. The second author [9] studies the case where γ : S n → R > is a C functionand proved that, if γ is convex (see § γ whose r -th anisotropic mean curvature for γ (Definition2.2) is integrable for r = 1 , · · · , n is (up to translation and homothety) the Wulff shape.However, the situation is not the same for non-convex γ . Actually in this paper we provethe following non-uniqueness results. 2 heorem 1.1. There exists a C ∞ function γ : S n → R > which is not a convex integrandsuch that there exist closed embedded CAMC hypersurfaces in R n +1 for γ each of which isnot (any homothety or translation of ) the Wulff shape. Theorem 1.2.
There exists a C ∞ function γ : S → R > which is not a convex integrandsuch that there exist closed embedded CAMC surfaces in R with genus zero for γ each ofwhich is not (any homothety or translation of ) the Wulff shape. These results are proved by giving examples ( §
3, 4, 5).The same examples will be applied to the anisotropic mean curvature flow. In order togive the precise statement, we recall that the Cahn-Hoffman map ξ γ for γ is 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 (cf.[9]). Let X t : M → R n +1 be one-parameter family of piecewise- C weak immersionswith anisotropic mean curvature Λ t . Assume that the Cahn-Hoffman field ˜ ξ t (which is ananisotropic generalization of the unit normal vector field. see § X t is defined on M . If X t satisfies ∂X t /∂t = Λ t ˜ ξ t , it is called an anisotropic mean curvatureflow, which diminishes the anisotropic energy if Λ t § § Theorem 1.3.
Let c be a positive constant. Set X t := p c − t ) ξ γ , t ≤ c. Then X t is a self-similar shrinking solution of the anisotropic mean curvature flow for γ ,that is(i) ∂X t /∂t = Λ t ˜ ξ t , and(ii) X t is homothetic to ξ γ and it shrinks as t increases. By using this result and by giving examples, we prove the following result ( §
3, 4, 6).
Theorem 1.4.
There exists a C ∞ function γ : S n → R > which is not a convex integrandsuch that there exist closed embedded self-similar shrinking solutions in R n +1 for γ each ofwhich is homeomorphic to S n and is not (any homothety or translation of ) the Wulff shape. In contrast with this result, the round sphere is the only closed embedded self-similarshrinking solution of mean curvature flow in R with genus zero ([4]).Finally we give two conjetures about the uniqueness problems studied in Theorems 1.1,1.2, and 1.4. Conjecture 1.1.
Assume that γ : S n → R > is of C . Let X : M → R n +1 be a closedCAMC hypersurface. We assume that the r -th anisotropic mean curvature of X for γ is ntegrable for r = 1 , · · · , n . Then, if X satisfies at least one of the conditions (I), (II), (III)above, X ( M ) is a subset of a homothety of the image ξ γ ( S n ) of the Cahn-Hoffman map ξ γ . Conjecture 1.2.
Assume that γ : S → R > is of C . Then any closed embedded self-similar shrinking solution of the anisotropic mean curvature flow for γ in R with genuszero is a subset of a homothety of ξ γ ( S ) . This article is organized as follows. § § n = 1 (Proposition 3.1). In § § §
6, we prove Theorems 1.3, 1.4. C r weak immersion and its anisotropic en-ergy First we recall the definition of a piecewise- C r weak immersion , ( r ∈ N ), defined in [9].Let M = ∪ ki =1 M i be an n -dimensional oriented compact connected C ∞ manifold, whereeach M i is an n -dimensional connected compact submanifold of M with 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 weak immersion (or a piecewise- C r weakly immersed hypersurface) if X satisfies the following conditions (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) The restriction X | M oi of X to the interior M oi of M i 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 , { 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 hypersurface. Denote by S ( X ) the set of all singular points of X , here asingular point of X is a point in M at which X is not an immersion.The anisotropic energy of a piecewise- C weak immersion X : M → R n +1 is definedas follows. 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 ) . The anisotropic energy F γ ( X ) of X is defined as F γ ( X ) := Z M γ ( ν ) dA := k X i =1 Z M i γ ( ν i ) dA. (1)4f 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 ). Definition 2.1.
Assume that S is a closed hypersurface in R n +1 that is the boundary of abounded connected 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 } holds.(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.A positive continuous function γ : S n → R > is called a convex integrand (or, simply,convex) if the set { γ ( ν ) − ν | ν ∈ S n } is convex. This condition is equivalent to the conditionthat the homogeneous extension γ : R n +1 → R ≥ of γ that is defined as γ ( rX ) := rγ ( X ) , ∀ X ∈ S n , ∀ r ≥ , (2)is a convex function (that is, γ ( X + Y ) ≤ γ ( X ) + γ ( Y ), ∀ X, Y ∈ R n +1 ) (cf. [15]).Now we assume that γ : S n → R > is a positive continuous function. Such a function issometimes called 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 called the Wulff shape for γ , where h , i stands for the standard innerproduct in R n +1 . We should remark that originally ˜ W [ γ ] itself was called the Wulff shape.However, in this paper, because we mainly study hypersurfaces, we call W γ := ∂ ˜ W [ γ ] theWulff shape. In the special case where γ ≡ W γ coincides with S n . W γ is not smooth in general. W γ is smooth and strictly convex if and only if γ is of C and the n × n matrix D γ + γ · S n , where D γ is theHessian of γ on S n and 1 is the identity matrix of size n . From now on, we assume that γ : S n → R > is a positive C function. We use the samenotations as in § ξ γ : S n → R n +1 (for γ ) is defined as ξ ( ν ) := ξ γ ( ν ) := Dγ + γ ( ν ) ν, ν ∈ S n , (3)here T ν ( S n ) for each ν ∈ S n is naturally identified with a hyperplane in R n +1 . The Wulffshape W γ is a subset of the image ˆ W γ := ξ γ ( S n ) of ξ γ . ˆ W γ = W γ holds if and only if γ isconvex, and in this case γ is the support function of W γ (cf. [9]).5 simple calculation shows that ξ γ is represented by using the homogeneous extension γ : R n +1 → R ≥ of γ as ξ γ ( ν ) = Dγ | ν , ν ∈ S n , (4)where D is the gradient in R n +1 .Let X : M = ∪ ki =1 M i → R n +1 be a piecewise- C weak immersion. The Cahn-Hoffmanfield ˜ ξ i along X i = X | M i for γ (or the anisotropic Gauss map of X for γ ) is defined as˜ ξ i := ξ γ ◦ ν i : M i → R n +1 . The linear map S γp : T p M i → T p M i given by the n × n matrix S γ := − d ˜ ξ i is called the anisotropic shape operatior of X i . Definition 2.2 (anisotropic curvatures, cf. [14], [5]) . (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. Take the unit normal vector ˜ ν of the Cahn-Hoffman map ξ so that ˜ ν = ν holds at any point ν ∈ S n where det A > { e , · · · , e n } bean orthonormal basis of T ν S n such that { ˜ ν, f , · · · , f n } is compatible with the canonicalorientation 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( ν ) is the anisotropic mean curvature of ξ at ν with respect to ˜ ν , thenΛ( ν ) = ( − , (det A ( ν ) > , , (det A ( ν ) < . (6) We give an example which will be used to prove Theorems 1.1, 1.2, and 1.4 in the followingsections. Throughout this section γ : S → R > is the function defined by γ ( e iθ ) := cos θ + sin θ, (7)where R is identified with C . Lemma 3.1. The Cahn-Hoffman map ξ γ : S → R for γ is represented as follows. ξ γ ( e iθ ) = (cid:0) (cos θ )(cos θ + 6 cos θ sin θ − θ ) , (sin θ )( − θ + 6 cos θ sin θ + sin θ ) (cid:1) . (8) In other words, we have ξ γ ( ν ) = (cid:0) ν ( ν + 6 ν ν − ν ) , ν ( − ν + 6 ν ν + ν ) (cid:1) , ν = ( ν , ν ) ∈ S . (9) Set ν = ± p − ν . Then ξ γ ( ν ) = − ν (9 ν − ν + 5) , ∓ q − ν (9 ν − ν − ! (10)= − (9 cos θ − 15 cos θ + 5) cos θ, − (9 cos θ − θ − 1) sin θ ! (11) holds.Proof. We use the formula (4), that is ξ γ ( ν ) = Dγ | ν , ν ∈ S n .γ is given by γ ( ν , ν ) = ν + ν ( ν + ν ) / . (12)Hence, we have γ ν = ν ( ν + 6 ν ν − ν )( ν + ν ) / , (13) γ ν = ν ( − ν + 6 ν ν + ν )( ν + ν ) / . (14)74) with (13) and (14) gives ξ γ ( ν ) = (cid:0) ν ( ν + 6 ν ν − ν ) , ν ( − ν + 6 ν ν + ν ) (cid:1) , ν = ( ν , ν ) ∈ S , (15)which gives the desired formulas (8), (9). Inserting ν = ± p − ν to (9), we obtain (10),(11). ✷ The image ξ γ ( S ) of the Cahn-Hoffman map ξ γ and the Wulff shape W γ are shown inFigure 1. Lemma 3.3 below gives rigorous explanations of these shapes. - - - - - - - - - - Figure 1: The image ξ γ ( S ) of the Cahn-Hoffman map (left) and the Wulff shape W γ (right)for γ defined by (7). W γ is a subset of ξ γ ( S ).Now we study the singular points of ξ γ . Lemma 3.2. (i) The the set S ( ξ γ ) of all singular points of ξ γ is given by S ( ξ γ ) = ( p := (cid:16) √ √ , √ − √ (cid:17) , p := (cid:16) √ − √ , √ √ (cid:17) ,p := (cid:16) − √ − √ , √ √ (cid:17) , p := (cid:16) − √ √ , √ − √ (cid:17) ,p := (cid:16) − √ √ , − √ − √ (cid:17) , p := (cid:16) − √ − √ , − √ √ (cid:17) ,p := (cid:16) √ − √ , − √ √ (cid:17) , p := (cid:16) √ √ , − √ − √ (cid:17)) . (16) (ii) Set p j = (cos θ j , sin θ j ) , j = 1 , · · · , , where θ is chosen so that < θ < π/ holds. Then, by choosing suitable principal valuesfor θ , · · · , θ , one can write θ = π/ − θ , θ = π/ θ , θ = π − θ , θ = π + θ , (17) θ = 3 π/ − θ , θ = 3 π/ θ , θ = − θ . (18) (iii) ξ γ ( p ) = ξ γ ( p ) = 2 √ , , ξ γ ( p ) = ξ γ ( p ) = 2 √ − , , (19) ξ γ ( p ) = ξ γ ( p ) = 2 √ − , − , ξ γ ( p ) = ξ γ ( p ) = 2 √ , − 1) (20)8 olds. Remark 3.1. Choose the principal value for θ so that 0 < θ < π/ θ ≈ . π. (21) Proof of Lemma 3.2. Using (10), we obtain ξ ′ γ ( ν ) = (cid:16) − ν − ν + 1) , ± ν p − ν (9 ν − ν + 1) (cid:17) . (22)Hence, ( ν , ν ) ∈ S is a singular point of ξ γ if and only if9 ν − ν + 1 = 0 (23)holds. (23) is equivalent to ν = √ ± √ , − √ ± √ , (24)which proves (i).(ii) is a consequence of (i).(iii) is obtained by inserting (16) to (10). ✷ Set ξ γ ( θ ) = (( ξ γ ) x ( θ ) , ( ξ γ ) y ( θ )) := ξ γ ( e iθ ) . Lemma 3.3. Increase and decrease of ( ξ γ ) x ( θ ) , ( ξ γ ) y ( θ ) , and the sign of A = D γ + γ · are given by the following table. θ θ θ θ π/ θ θ ( ξ γ ) ′ x ( θ ) 0 − − − ξ γ ) ′ y ( θ ) 0 − − − − − ξ γ ) x ( θ ) 2 / √ ց ր / √ ց − / √ ր ր / √ ց − / √ ξ γ ) y ( θ ) 2 / √ ց ց − / √ ր / √ ց ր / √ ց − / √ A = D γ + γ · − − − − − − θ θ π θ θ π/ θ θ + 2 π ( ξ γ ) ′ x ( θ ) 0 + 0 − − − − ξ γ ) ′ y ( θ ) 0 + 5 + 0 − − ξ γ ) x ( θ ) − / √ ր − ց − / √ ր / √ ց ց − / √ ր / √ ξ γ ) y ( θ ) − / √ ր ր / √ ց − / √ ր − ց − / √ ր / √ A = D γ + γ · − − − − − − roof. Using (11), we obtain ξ ′ γ ( θ ) = (( ξ γ ) ′ x ( θ ) , ( ξ γ ) ′ y ( θ )) = (cid:16) θ − θ +1) sin θ, − θ − θ +1) cos θ (cid:17) . (25)For γ ( θ ) := γ ( e iθ ) = cos θ + sin θ , we compute γ ′ ( θ ) = − θ sin θ + 6 sin θ cos θ,γ ′′ ( θ ) = 6(5 cos θ sin θ − cos θ − sin θ ) , and hence A = γ ′′ ( θ ) + γ ( θ ) = − θ − θ + 1) = − ( ξ γ ) ′ x ( θ ) · (sin θ ) − (26)holds, here the last equality is valid only when sin θ = 0. These observations with Lemmas3.1, 3.2 give the desired result. ✷ Using Lemma 3.1, by simple calculations, we obtain the following result. Lemma 3.4. Let ρ , ρ ∈ (0 , π/ be the solutions of cos ρ = s √ , cos ρ = s − √ , (27) respectively. Then(i) < θ < ρ < π/ < ρ < θ < π/ holds.(ii) Set α := s √ 53 ( √ − . (29) The “inner self-intersection points” of ξ γ are the following four points. Q := ( α, 0) = ξ γ ( ρ ) = ξ γ ( ρ + 3 π/ , Q := (0 , α ) = ξ γ ( ρ ) = ξ γ ( ρ + π/ ,Q := ( − α, 0) = ξ γ ( ρ + π ) = ξ γ ( ρ + π/ , Q := (0 , − α ) = ξ γ ( ρ + π ) = ξ γ ( ρ + 3 π/ .Q , Q , Q , Q are the vertices of W γ (Figure 1, right). Remark 3.2. By computation using Mathematica ver.11.2.0.0, we get ρ ≈ . π, ρ = π/ − ρ ≈ . π, α ≈ . . (30)By using the previous lemmas 3.1, 3.2, 3.3, and 3.4 with Fact 2.2 (i), we obtain thefollowing: Proposition 3.1. Let γ : S → R > be the function defined by γ ( e iθ ) := cos θ + sin θ. (31)10 et θ , θ be the constants defined in Lemma 3.2, and let ρ , ρ be the constants defined inLemma3.4.(i) The Cahn-Hoffman map ξ γ : S → R is represented as ξ γ ( e iθ ) = (cid:0) (cos θ )(cos θ + 6 cos θ sin θ − θ ) , (sin θ )( − θ + 6 cos θ sin θ + sin θ ) (cid:1) (32) (Figure 1).(ii) The Wulff shape W γ is given by W γ = ξ γ (cid:16) { ρ ≤ θ ≤ ρ } ∪ { ρ + π/ ≤ θ ≤ ρ + π/ }∪{ ρ + π ≤ θ ≤ ρ + π } ∪ { ρ + 3 π/ ≤ θ ≤ ρ + 3 π/ } (cid:17) (33) (Figure 2a).(iii) The following four closed curves which are subsets of ξ γ ( S ) are closed piecewise- C ∞ CAMC curves (Figures 2b, 2c, 2d, 2e). ( C γ ) := ξ γ (cid:16) { θ ≤ θ ≤ θ } ∪ { θ ≤ θ ≤ θ }∪{ θ ≤ θ ≤ θ } ∪ { θ ≤ θ ≤ θ } (cid:17) , (34)( C γ ) := ξ γ (cid:16) { θ ≤ θ ≤ θ } ∪ { θ ≤ θ ≤ θ } (cid:17) , (35)( C γ ) := ξ γ (cid:16) { θ ≤ θ ≤ θ } ∪ { θ ≤ θ ≤ ρ + π/ } ∪ { ρ + π ≤ θ ≤ θ } (cid:17) , (36)( C γ ) := ξ γ (cid:16) { θ ≤ θ ≤ ρ } ∪ { ρ ≤ θ ≤ θ }∪{ θ ≤ θ ≤ ρ + π/ } ∪ { ρ + π/ ≤ θ ≤ θ } (cid:17) ∪{ θ ≤ θ ≤ ρ + π } ∪ { ρ + π ≤ θ ≤ θ } (cid:17) ∪{ θ ≤ θ ≤ ρ + 3 π/ } ∪ { ρ + 3 π/ ≤ θ ≤ θ + 2 π } (cid:17) . (37) The anisotropic (mean) curvature for the outward-pointing normal is − .(iv) The following closed curve which is a subset of ξ γ ( S ) is a closed piecewise- C ∞ curve (Figure 2f). ( C γ ) := ξ γ (cid:16) {− ρ ≤ θ ≤ ρ } (cid:17) . (38) Its anisotropic (mean) curvature is not constant, because, for the outward-pointing normal,it is − at each point in the solid curve: ξ γ (cid:16) { θ ≤ θ ≤ θ } (cid:17) , (39) while it is at each point in the dashed curves: ξ γ (cid:16) {− ρ ≤ θ ≤ θ } ∪ { θ ≤ θ ≤ ρ } (cid:17) . (40)11 a) Wulff shape W γ (b) CAMC curve ( C γ ) (c) CAMC curve ( C γ ) (d) CAMC curve ( C γ ) (e) CAMC curve ( C γ ) (f) curve ( C γ ) . not CAMC Figure 2: Some of the closed curves which are subsets of ξ γ ( S ) for γ defined by (7) (cf.Figure 1). (a), (b), (c), (d), (e) : The anisotropic (mean) curvature for the outward-pointingnormal is − 1. (f) : For the outward-pointing normal, the anisotropic (mean) curvature is − Remark 3.3. Proposition 3.1 proves Theorem 1.1 for n = 1. Remark 3.4. In Proposition 3.1, we gave six closed piecewise- C ∞ curves which are subsetsof ξ γ ( S ). Five of them were CAMC and the other one was not CAMC. We should remarkthat there are more piecewise- C ∞ CAMC closed curves and non-CAMC closed curves.Figure 3 gives all of the other closed CAMC curves included in ξ γ ( S ) (up to congruencein R ). 12 a) CAMC curve (b) CAMC curve (c) CAMC curve(d) CAMC curve (e) CAMC curve (f) CAMC curve(g) CAMC curve (h) CAMC curve Figure 3: Closed CAMC curves which are subsets of ξ γ ( S ) for γ defined by (7) (cf. Figure1). For all of these eight curves, the anisotropic (mean) curvature for the outward-pointingnormal is − 1. They and the five curves in Figures 2a, 2b, 2c, 2d, 2e give all closed CAMCcurves included in ξ γ ( S ) (up to congruence in R ). In this section we give two higher dimensional examples by rotating γ which was definedby (7) and studied in § γ ( ν , ν ) = ν + ν ( ν + ν ) / defined on the ( ν , ν )-plane. We denote the restriction of γ to S by γ . Then, by Lemma3.1, the Cahn-Hoffman map ξ γ : S → R for γ is represented as follows (Figure 4a). ξ γ ( ν ) = (cid:0) ν ( ν + 6 ν ν − ν ) , ν ( − ν + 6 ν ν + ν ) (cid:1) (41)= (cid:0) (cos θ )(cos θ + 6 cos θ sin θ − θ ) , (sin θ )( − θ + 6 cos θ sin θ + sin θ ) (cid:1) . (42)( ν = ( ν , ν ) = (cos θ, sin θ ) ∈ S ). The higher dimensional example obtained by rotating13 around the ν -axis is given by γ ( ν , ν , ν ) = ( ν + ν ) + ν ( ν + ν + ν ) / , ( ν , ν , ν ) ∈ R . (43)The corresponding Cahn-Hoffman map ξ γ : S → R is given as follows (Figure 4b). ξ γ ( ν ) = (cid:0) (cos θ )(cos θ + 6 cos θ sin θ − θ )(cos ρ ) , (cos θ )(cos θ + 6 cos θ sin θ − θ )(sin ρ ) , (sin θ )( − θ + 6 cos θ sin θ + sin θ ) (cid:1) , (44)( ν = (cos θ cos ρ, cos θ sin ρ, sin θ ) ∈ S ).By the same way as in § 3, we get closed piecewise- C ∞ CAMC surfaces and closedpiecewise- C ∞ non-CAMC surfaces for γ which are subsets of ξ γ ( S ) and which are notthe Wulff shape W γ (up to homothety and translation). In fact, we have the following: Proposition 4.1. Consider γ : S → R > defined by γ ( ν , ν , ν ) = ( ν + ν ) + ν , ( ν , ν , ν ) ∈ S . (45) The Wulff shape W γ is the surface of revolution (Figure 5a) given by rotating W γ (Figure2a) around the vertical axis. The two piecewise- C ∞ closed surfaces (Figures 5b, 5c) givenby rotating the closed curves ( C γ ) , ( C γ ) (Figures 2b, 2e) around the vertical axis areCAMC. The piecewise- C ∞ closed surface (Figures 5d) given by rotating the closed curve ( C γ ) (Figure 2f) is not CAMC. - - - - (a) The image of ξ γ (b) The image of ξ γ Figure 4: (a): The image of the Cahn-Hoffman map ξ γ : S → R for γ : S → R > . (b):The image of the Cahn-Hoffman map ξ γ : S → R for γ : S → R > .14 a) Wulff shape W γ (b) CAMC surface(c) CAMC surface (d) not CAMC Figure 5: Some of the closed surfaces which are subsets of ξ γ ( S ) for γ defined by (45)(Figure 4b). They are surfaces given by rotating the curves W γ , ( C γ ) , ( C γ ) , ( C γ ) ,respectively. (a), (b), (c): The anisotropic mean curvature for the outward-pointing normalis − 1. (d) : The anisotropic mean curvature is − γ around the origin by π/ γ ( ν , ν ) = ν + 15 ν ν + 15 ν ν + ν ν + ν ) / (46)defined on the ( ν , ν )-plane. We denote the restriction of γ to S by γ . The Cahn-Hoffman map ξ γ : S → R for γ is obtained by rotating ξ γ around the origin by π/ ξ γ ( ν ) = 14 (cid:0) ν ( ν − ν ν + 15 ν ν + 25 ν ) , ν (25 ν + 15 ν ν − ν ν + ν ) (cid:1) (47)= 14 (cid:0) (cos θ )(cos θ − θ sin θ + 15 cos θ sin θ + 25 sin θ ) , (sin θ )(25 cos θ + 15 cos θ sin θ − θ sin θ + sin θ ) (cid:1) . (48)( ν = ( ν , ν ) = (cos θ, sin θ ) ∈ S ). The higher dimensional example obtained by rotating γ around the ν -axis is given by γ ( ν , ν , ν ) = ( ν + ν ) + 15( ν + ν ) ν + 15( ν + ν ) ν + ν ν + ν + ν ) / , ( ν , ν , ν ) ∈ R . (49)The restriction γ of γ : R → R ≥ to S can be written as γ ( ν , ν , ν ) = ( ν + ν ) + 15( ν + ν ) ν + 15( ν + ν ) ν + ν , ( ν , ν , ν ) ∈ S . (50)15he corresponding Cahn-Hoffman map ξ γ : S → R is given as follows (Figure 6b). ξ γ ( ν ) = 14 (cid:0) (cos θ )(cos θ − θ sin θ + 15 cos θ sin θ + 25 sin θ )(cos ρ ) , (cos θ )(cos θ − θ sin θ + 15 cos θ sin θ + 25 sin θ )(sin ρ ) , (sin θ )(25 cos θ + 15 cos θ sin θ − θ sin θ + sin θ ) (cid:1) , (51)( ν = (cos θ cos ρ, cos θ sin ρ, sin θ ) ∈ S ).By the same way as above, we get closed piecewise- C ∞ CAMC surfaces (Figure 7) andclosed piecewise- C ∞ non-CAMC surfaces for γ which are subsets of ξ γ ( S ) and which arenot the Wulff shape W γ (up to homothety and translation). - - - - - - (a) The image of ξ γ (b) The image of ξ γ Figure 6: (a): The image of the Cahn-Hoffman map ξ γ : S → R for γ : S → R > . (b):The image of the Cahn-Hoffman map ξ γ : S → R for γ : S → R > . (a) Wulff shape W γ (b) CAMC surface for γ (c) CAMC surface for γ Figure 7: Some of the closed surfaces which are subsets of ξ γ ( S ) for γ defined by (50)(Figure 6b). The anisotropic mean curvature for the outward-pointing normal is − The functions γ : S → R > , γ : S → R > and their rotations around the vertical axisgive examples which prove Theorems 1.1, 1.2. In fact, Propositions 3.1, 4.1 give suitableexamples for n = 1 , 2, respectively. Also, higher dimensional examples are obtained by themethod given in § 4. 16 Applications to anisotropic mean curvature flow: proofs ofTheorems 1.3, 1.4 Let γ : S n → R > be of C with Cahn-Hoffman map ξ γ . Let X t : M → R n +1 be one-parameter family of embedded piecewise- C hypersurfaces with anisotropic mean curva-ture Λ t . Assume that the Cahn-Hoffman field ˜ ξ t along X t is defined on M . If X t satis-fies ∂X t /∂t = Λ t ˜ ξ t , it is called an anisotropic mean curvature flow, which diminishes theanisotropic energy if Λ t 0. In fact, d F γ ( X t ) dt = − Z M n Λ t D ∂X t ∂t , ν t E dA t = − Z M n Λ t h Dγ + γ ( ν t ) ν t , ν t i dA t = − Z M n Λ t γ ( ν t ) dA t ≤ Proof of Theorem 1.3. Since the anisotropic mean curvature of ξ γ is − 1, Λ t = − p c − t )holds. On the other hand, ˜ ξ t = ξ γ holds. These two facts imply that (i) and (ii) hold. ✷ Proof of Theorem 1.4. Examples stated in Propositions 3.1, 4.1 give the desired result. ✷ References [1] A. D. Alexandrov. A characteristic property of spheres. Ann. Mat. Pura Appl. (1962), 303–315.[2] N. Ando. Hartman-Wintner’s theorem and its applications. Calc. Var. Partial Differ-ential Equations (2012), 389–402.[3] J. L. Barbosa and M. do Carmo. Stability of hypersurfaces with constant mean cur-vature. Math. Z. (1984), 339–353.[4] S. Brendle. Embedded self-similar shrinkers of genus 0. Ann. of Math. (2016),715–728.[5] Y. He, H. Li, H. Ma and J. Ge. Compact embedded hypersurfaces with constant higherorder anisotropic mean curvatures. Indiana Univ. Math. J. (2009), 853-868.[6] H. Hopf. Differential geometry in the large. Lecture Notes in Math. . Springer-Verlag, Berlin, 1989.[7] N. Kapouleas. 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