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

Xu Cheng, Tito Mejia and Detang Zhou

Vol. 271 (2014), No. 2, 347–367

DOI: 10.2140/pjm.2014.271.347

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 on 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 uniform upper bounds on 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.

Keywords

Riemannian manifold, eigenvalue, drifted Laplacian, minimal surface

Mathematical Subject Classification 2010

Primary: 58J50

Secondary: 58E30

Milestones

Received: 4 July 2013

Revised: 14 May 2014

Accepted: 30 July 2014

Published: 20 September 2014