Illinois Journal of Mathematics

Stability of hypersurfaces with constant $(r+1)$-th anisotropic mean curvature

Yijun He and Haizhong Li

Full-text: Open access


Given a positive function $F$ on $S^n$ which satisfies a convexity condition, 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. Let $X : M\to\mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=\mathrm{constant}$, for some $r$ with $0\leq r\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape.

Article information

Illinois J. Math., Volume 52, Number 4 (2008), 1301-1314.

First available in Project Euclid: 18 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]


He, Yijun; Li, Haizhong. Stability of hypersurfaces with constant $(r+1)$-th anisotropic mean curvature. Illinois J. Math. 52 (2008), no. 4, 1301--1314. doi:10.1215/ijm/1258554364.

Export citation


  • H. Alencar, M. do Carmo and H. Rosenberg, On the first eigenvalue of the linearized operator of the $r$-th mean curvature of a hypersurface, Ann. Global Anal. Geom. 11 (1993), 387–395.
  • J. L. M. Barbosa and A. G. Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), 277–297.
  • L. F. Cao and H. Li, $r$-minimal submanifolds in space forms, Ann. Global Anal. Geom. 32 (2007), 311–341.
  • U. Clarenz, The Wulff-shape minimizes an anisotropicWillmore functional, Interfaces Free Bound. 6 (2004), 351–359.
  • L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965.
  • G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
  • Y. J. He and H. Li, Integral formula of Minkowski type and new characterization of the Wulff shape, Acta Math. Sin. (Engl. Ser.) 24 (2008), 697–704.
  • Y. J. He and H. Li, A new variational characterization of the Wulff shape, Differential Geom. App. 26 (2008), 377–390.
  • Y. J. He, H. Li, H. Ma and J. Q. Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. Math. J. 58 (2009), 853–868.
  • Z. J. Hu and H. Li, Willmore submanifolds in Riemannian manifolds, Proceedings of the Workshop Contem. Geom. and Related Topics (Belgrad, Yugoslavia, May, 2002) 15–21, World Scientific, 2002, pp. 251–275.
  • M. Koiso and B. Palmer, Geometry and stablity of surfaces with constant anisotropic mean curvature, Indiana Univ. Math. J. 54 (2005), 1817–1852.
  • H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), 665–672.
  • H. Li, Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), 327–351.
  • B. Palmer, Stability of the Wulff shape, Proc. Amer. Math. Soc. 126 (1998), 3661–3667.
  • R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom. 8 (1973), 465–477.
  • R. Reilly, The relative differential geometry of nonparametric hypersurfaces, Duke Math. J. 43 (1976), 705–721.
  • H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Soc. Math. 117 (1993), 211–239.
  • J. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), 568–588.
  • S. Winklmann, A note on the stability of the Wulff shape, Arch. Math. 87 (2006), 272–279.
  • K. Yano, Integral formulas in Riemannian geometry, Dekker, New York, 1970.