Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space. (English) Zbl 1395.53070

Let \(F:\mathbb{S}^{n}\longrightarrow\mathbb{R}^{+}\) be a positive smooth function which satisfies the following convexity condition: \[ (D^{2}F + F\mathbb{I})_{x}>0,\forall\, x\in \mathbb{S}^{n}, \] where \(D^{2} F\) denotes the intrinsic Hessian of \(F\) on the \(n\)-sphere \(\mathbb{S}^{n}\) of \(\mathbb{R}^{n+1}\), \(\mathbb{I}\) denotes the identity on \(T_{x}\mathbb{S}^{n}\), and \(>0\) means that the matrix is positive definite. The authors consider the map \[ \phi: \mathbb{S}^{n}\longrightarrow \mathbb{R}^{n+1},\;\; x\longmapsto F(x)x+(\nabla_{\mathbb{S}^{n}}F)_{x}, \] whose image \(\mathcal{W}_{F} = \phi(\mathbb{S}^{n})\) is a smooth, convex hypersurfaces in \(\mathbb{R}^{n+1}\) called the Wulff shape of \(F\). Let \(r\) and \(s\) be integers satisfying \(0\leq r\leq s\leq n-2\), \(n\geq 3\), and let \(x: M^{n}\hookrightarrow\mathbb{R}^{n+1}\) be a closed (\(r, s, F )\)-linear Weingarten hypersurface with \(H_{s+1}^{F}\) positive. Then, they prove that the smooth immersion \(x: M^{n}\hookrightarrow\mathbb{R}^{n+1}\) is stable if and only if, up to translations and homotheties, \(x(M )\) is the Wulff shape of \(F\).


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds
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