$f$-minimal surface and manifold with positive $m$-Bakry-Émery Ricci curvature

Abstract

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-Émery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-Émery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.

15 pages

References in corpus (3)

• Volume growth, eigenvalue and compactness for self-shrinkers
• Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary
• On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking Ricci solitons

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• Applications of Some Elliptic Equations in Riemannian Manifolds
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• Sharp gradient estimates on weighted manifolds with compact boundary
• The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces
• Stability properties and gap theorem for complete $f$-minimal hypersurfaces
• Bakry-Émery Ricci Curvature Bounds on Manifolds with Boundary
• Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds $N\times \mathbb{R}$
• Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces
• The curve shortening problem associated to a density