Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space (Q1704356)

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 \(F\).