Upper bounds for the first stability eigenvalue of surfaces in 3-Riemannian manifolds
type: Item
instance of: scholarly article
title: Upper bounds for the first stability eigenvalue of surfaces in 3-Riemannian manifolds
zbMATH Open document ID: 1401.53033
author: Márcio Batista
author: José I. Santos
published in: Potential Analysis
publication date: 2018-07-13T00:00:00Z
review text: 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.
reviewed by: Loreno Heer
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 6903323
zbMATH Keywords: eigenvalues
zbMATH Keywords: weighted mean curvature
zbMATH Keywords: closed surfaces
MaRDI profile type: Publication
full work available at URL: s11118-017-9649-3
OpenAlex ID: W2753218917
Wikidata QID: Q115381397
cites work: A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue
cites work: A spectral characterization of the $H(r)$-torus by the first stability eigenvalue
cites work: On the first stability eigenvalue of constant mean curvature surfaces into homogeneous 3-manifolds
cites work: Q5186515
cites work: Conformal invariants associated to a measure
cites work: Stability and compactness for complete 𝑓-minimal surfaces
cites work: Topology of three-manifolds with positive 𝑃-scalar curvature
cites work: Isoperimetry of waists and concentration of maps
cites work: Stability properties and topology at infinity of \(f\)-minimal hypersurfaces
cites work: Q5609089
cites work: Q2550561
cites work: Q5312205
cites work: Analysis of weighted Laplacian and applications to Ricci solitons
cites work: Geometry of manifolds with densities
cites work: First stability eigenvalue characterization of Clifford hypersurfaces
cites work: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature
cites work: Complete Riemannian Manifolds and Some Vector Fields
cites work: Optimal Transport
cites work: Comparison geometry for the Bakry-Emery Ricci tensor
cites work: Sectional curvature for Riemannian manifolds with density
DOI: 10.1007/S11118-017-9649-3
description: scientific article; zbMATH DE number 6903323