Upper bounds for the first stability eigenvalue of surfaces in 3-Riemannian manifolds
A weighted manifold is a Riemannian manifold \((M^3, g)\) with a real-valued smooth function \(f:M \to \mathbb{R}\) which is used as density. Associated to this structure is the Drift Laplacian \(\Delta_f u = \Delta u = \langle\nabla u, \nabla f\rangle\). Let \(S_\infty = S + 2 \Delta_M f - |\bar\nabla f |^2\) be the Perelman scalar curvature. Furthermore the weighted mean curvature is given by \(H_f = H + \langle N, \bar\nabla f \rangle\). The authors give the following theorem: ``Let \((M^3, g, f)\) be a weighted manifold with \(S_\infty \geq 6c\), for some \(c \in \mathbb{R}\). Let \(\Sigma^2 \subset M^3\) be a closed surface with constant weighted mean curvature \(H_f\). Then: \[ \lambda_1 \leq - \frac{1}{2}(H_f^2 +6c) - \frac{4 \pi(g-1)}{|\Sigma|}. \] Moreover, equality holds if and only if \(\Sigma\) is totally geodesic, \(f\) is constant on \(\Sigma\), \({S_{\infty}}_{|\Sigma} = 6c\) and \(K\) is constant.'' Here, \(\lambda_1\) denotes the first eigenvalue of the Jacobi operator. Two more theorems are given.
scientific article; zbMATH DE number 6903323
scientific article