## 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 $$F$$.

### MSC:

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