COMPACT EMBEDDED HYPERSURFACES WITH CONSTANT HIGHER ORDER ANISOTROPIC MEAN CURVATURES

By Yijun He, Haizhong Li, Hui Ma, Jianquan Ge. 2007

93 citations recorded by the source corpus.

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\R^{n+1}$ with $H^F_r={constant}$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5e354cbb8fd2dd0ffb69270c2db26a5c81ad54f8

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In Applied Mathematics

1. Giulio Ciraolo
2. Mi-Ho Giga
3. Yoshikazu Giga
4. C. Bianchini

In Geometry and Topology

1. Giulio Ciraolo
2. Mi-Ho Giga
3. Yoshikazu Giga

Citations in the corpus

• Wulff shape characterizations in overdetermined anisotropic elliptic problems (2017, result): “As shown in [9] and [30], an analogous result holds in the Finsler metric.”
• Very singular diffusion equations: second and fourth order problems (2010, background): “For development of this subject for strictly convex smooth energy the reader is referred to [ 67 ,81,82] and references cited there.”