Eigenvalue estimate and compactness for closed $f$-minimal surfaces

Abstract

Let $\Omega$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded $f$-minimal hypersurfaces contained in $\Omega$. Using this estimate, we prove a compactness theorem for the space of closed embedded $f$-minimal surfaces with the uniform upper bounds of genus and diameter in a complete $3$-manifold with Bakry-Émery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary.

25 pages

References in corpus (3)

• $f$-minimal surface and manifold with positive $m$-Bakry-Émery Ricci curvature
• Volume growth, eigenvalue and compactness for self-shrinkers
• Stability and compactness for complete $f$-minimal surfaces

Cited by in corpus (4)

• Eigenvalues of the drifted Laplacian on complete metric measure spaces
• Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor
• Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition
• Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds