# Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

## Abstract

**:**

## 1. Introduction

**Question**

**1.**Is any closed CAMC hypersurface 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 ${S}^{2}$.

**Theorem**

**1**

**.**There exists a ${C}^{\infty}$ function $\gamma :{S}^{n}\to {\mathbb{R}}_{>0}$ such that there exist closed embedded CAMC hypersurfaces in ${\mathbb{R}}^{n+1}$ for γ, each of which is not (any homothety or translation of) the Wulff shape ${W}_{\gamma}$.

**Theorem**

**2**

**.**There exists a ${C}^{\infty}$ function $\gamma :{S}^{2}\to {\mathbb{R}}_{>0}$ such that there exist closed embedded CAMC surfaces in ${\mathbb{R}}^{3}$ with genus zero for γ, each of which is not (any homothety or translation of) the Wulff shape ${W}_{\gamma}$.

**Theorem**

**3**

**.**Assume that $\gamma :{S}^{n}\to {\mathbb{R}}_{>0}$ is of class ${C}^{2}$ and convex. Then, the image of any closed stable piecewise-${C}^{2}$ CAMC hypersurface for γ of which the rth anisotropic mean curvature for γ (see Section 3) is integrable for $r=1,\dots ,n$ is (up to translation and homothety) a covering of the Wulff shape ${W}_{\gamma}$.

**Theorem**

**4.**

## 2. Preliminaries

**Definition**

**1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3**

**.**We give a simple example, which shows that, if the energy density function γ is not strictly convex, even if it is convex, the Wulff shape can have singular points and its principal curvatures can be unbounded. Set $n=1$. For $m\in \mathbb{N}$, define

- (i)
- If $m\ge 2$, ${A}_{m}=0$ on $S\left[\gamma \right]:=\left\{\right(cos\theta ,sin\theta )\phantom{\rule{0.222222em}{0ex}};\phantom{\rule{0.222222em}{0ex}}\theta =(1/2)\ell \pi ,(\ell \in \mathbb{Z}\left)\right\}$.
- (ii)
- ${A}_{m}$ is positive definite on ${S}^{1}\setminus S\left[\gamma \right]$ and positive semi-definite on ${S}^{1}$.

**Example**

**4.**

## 3. Euler–Lagrange Equations and Anisotropic Curvatures

- (A1)
- X is continuous, and each ${X}_{i}{:=X|}_{{M}_{i}}:{M}_{i}\to {\mathbb{R}}^{n+1}$ is of class ${C}^{2}$.
- (A2)
- The restriction ${X|}_{{M}_{i}^{o}}$ of X to the interior ${M}_{i}^{o}$ of ${M}_{i}$ is a ${C}^{2}$-immersion.
- (A3)
- The unit normal vector field ${\nu}_{i}:{M}_{i}^{o}\to {S}^{n}$ along ${X}_{i}{|}_{{M}_{i}^{o}}$ can be extended to a ${C}^{1}$-mapping ${\nu}_{i}:{M}_{i}\to {S}^{n}$. Here, the orientation of ${\nu}_{i}$ is determined so that, if $({u}^{1},\dots ,{u}^{n})$ is a local coordinate system in ${M}_{i}$, then $\{{\nu}_{i},\partial /\partial {u}^{1},\dots ,\partial /\partial {u}^{n}\}$ gives the canonical orientation in ${\mathbb{R}}^{n+1}$.

**Definition**

**2**

**.**(i) The eigenvalues of ${S}^{\gamma}$ are called the anisotropic principal curvatures of X. We denote them by ${k}_{1}^{\gamma},\dots ,{k}_{n}^{\gamma}$.

**Remark**

**1**

**.**For the Cahn–Hoffman map ${\xi}_{\gamma}:{S}^{n}\to {\mathbb{R}}^{n+1}$, ${\xi}_{\gamma}{\left(\nu \right)=D\gamma |}_{\nu}+\gamma \left(\nu \right)\nu $, ($\nu \in {S}^{n}$), it is shown that the unit normal vector field ${\nu}_{{\xi}_{\gamma}}$ is given by ${\xi}_{\gamma}^{-1}$. Hence, the anisotropic shape operator of ${\xi}_{\gamma}$ is ${S}^{\gamma}=-d({\xi}_{\gamma}\circ {\nu}_{{\xi}_{\gamma}})=-d\left({\mathrm{id}}_{{S}^{n}}\right)=-{I}_{n}$. Therefore, the anisotropic principal curvatures of ${\xi}_{\gamma}$ are $-1$, and hence, each rth anisotropic mean curvature of ${\xi}_{\gamma}$ is ${(-1)}^{r}$. Particularly, the anisotropic mean curvature of ${\xi}_{\gamma}$ for the normal ν and that of ${W}_{\gamma}$ for the outward-pointing unit normal is $-1$ at any regular point.

**Remark**

**2**

**Proposition**

**1**

**.**Assume that the map $X:{M}_{0}\to {\mathbb{R}}^{n+1}$ satisfies (A1), (A2), and (A3) above with ${X}_{i}=X$, ${M}_{i}={M}_{0}$, and ${\nu}_{i}=\nu $. Let ${X}_{\u03f5}:{M}_{0}\to {\mathbb{R}}^{n+1}$ ($\u03f5\in J:=(-{\u03f5}_{0},{\u03f5}_{0})$), be a variation of X; that is, ${\u03f5}_{0}>0$ and ${X}_{0}=X$. Assume for simplicity that ${X}_{\u03f5}$ is of class ${C}^{\infty}$ in ϵ. We also assume that, for each $\u03f5\in J$, the anisotropic mean curvature ${\Lambda}_{\u03f5}$ of ${X}_{\u03f5}$ ($\u03f5\ne 0$) is bounded on ${M}_{0}^{o}$. Set

**Proposition**

**2**

**.**For $n=2$, see Palmer [9]). A piecewise-${C}^{2}$ weak immersion $X:M={\cup}_{i=1}^{k}{M}_{i}\to {\mathbb{R}}^{n+1}$ is a critical point of the anisotropic energy ${\mathcal{F}}_{\gamma}$ for volume-preserving variations if and only if the following conditions (i) and (ii) hold.

**Definition**

**3**

**.**A piecewise-${C}^{2}$ weak immersion $X:M={\cup}_{i=1}^{k}{M}_{i}\to {\mathbb{R}}^{n+1}$ is called a hypersurface with constant anisotropic mean curvature (CAMC) if both conditions (i) and (ii) in Proposition 2 hold.

## 4. Outline of the Proofs of Theorems 1 and 2

## 5. Outline of the Proof of Theorem 3

**Definition**

**4**

**.**Let X be a piecewise-${C}^{2}$ weak immersion. For any real number t, we call the map ${X}_{t}:=X+t\tilde{\xi}:M\to {\mathbb{R}}^{n+1}$ the anisotropic parallel deformation of X of height t. If ${X}_{t}$ is a piecewise-${C}^{2}$ weak immersion, then we call it the anisotropic parallel hypersurface of X of height t.

**Theorem**

**5**

**.**Assume that $\gamma :{S}^{n}\to {\mathbb{R}}_{>0}$ is of class ${C}^{2}$. Let $X:M={\cup}_{i=1}^{k}{M}_{i}\to {\mathbb{R}}^{n+1}$ be a piecewise-${C}^{2}$ weak immersion. Consider anisotropic parallel hypersurfaces ${X}_{t}=X+t\tilde{\xi}:M\setminus S\left[X\right]\to {\mathbb{R}}^{n+1}$, where $S\left[X\right]$ is the set of singular points of X. Then, the following integral formula holds.

**Theorem**

**6**

**.**Assume that $\gamma :{S}^{n}\to {\mathbb{R}}_{>0}$ is of class ${C}^{2}$. Assume also that $X:M={\cup}_{i=1}^{k}{M}_{i}\to {\mathbb{R}}^{n+1}$ is a closed piecewise-${C}^{2}$ weak immersion and that X satisfies the following condition.

## 6. Application to Anisotropic Mean Curvature Flow

## 7. Proof of Theorem 4

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**On the left, ${W}_{{\gamma}_{m}}$ for $m=1$ and, on the right, ${W}_{{\gamma}_{m}}$ for $m=2$ for ${\gamma}_{m}$ defined by Equation (2).

**Figure 2.**The image of the Cahn–Hoffman map ${\xi}_{\gamma}$ for γ in Example 4: The six vertices are the image of the singular points of ${\xi}_{\gamma}$, and the closed convex solid curve is the Wulff shape ${W}_{\gamma}$.

**Figure 3.**

**Left**: The image ${\xi}_{\gamma}\left({S}^{2}\right)$ of the Cahn–Hoffman map ${\xi}_{\gamma}:{S}^{2}\to {\mathbb{R}}^{3}$ for $\gamma :{S}^{2}\to {\mathbb{R}}_{>0}$ defined by Equation (6).

**Right**: The section of ${\xi}_{\gamma}\left({S}^{2}\right)$ in the $({x}_{1},{x}_{3})$-plane.

**Figure 4.**Some of the closed surfaces which are subsets of ${\xi}_{\gamma}\left({S}^{2}\right)$ for $\gamma $ defined by Equation (6) (Figure 3). The anisotropic mean curvature for the outward-pointing normal is $-1$: (

**a**) Wulff shape ${W}_{\gamma}$; (

**b**) a constant anisotropic mean curvature (CAMC) surface for $\gamma $; and (

**c**) a CAMC surface for $\gamma $.

**Figure 6.**Construction of volume-preserving variation ${Y}_{t}$ using anisotropic parallel surfaces ${X}_{t}$ of X.

**Upper left:**A capillary surface X and its anisotropic parallel surface ${X}_{t}$.

**Upper right:**A parallel translation ${Z}_{t}$ of ${X}_{t}$ that satisfies the boundary condition.

**Bottom:**A homothety ${Y}_{t}$ of ${Z}_{t}$ that satisfies the volume condition.

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**MDPI and ACS Style**

Koiso, M. Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem. *Math. Comput. Appl.* **2019**, *24*, 88.
https://doi.org/10.3390/mca24040088

**AMA Style**

Koiso M. Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem. *Mathematical and Computational Applications*. 2019; 24(4):88.
https://doi.org/10.3390/mca24040088

**Chicago/Turabian Style**

Koiso, Miyuki. 2019. "Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem" *Mathematical and Computational Applications* 24, no. 4: 88.
https://doi.org/10.3390/mca24040088