First eigenvalue of the $p$-Laplace operator along the Ricci flow

Abstract

In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative.

28 pages, added statements and references, deleted verbose statements and corrected typos

References in corpus (2)

• A Class of Monotonic Quantities along the Ricci Flow
• A Comparison Theorem and A Sharp Bound via the Ricci Flow

Cited by in corpus (8)

• Sharp estimate on the first eigenvalue of the p-Laplacian on compact manifold with nonnegative Ricci curvature
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Sharp Estimates for the Principal Eigenvalue of the p-Operator
• Eigenvalues under the Ricci flow of model geometries
• Kähler non-collapsing, eigenvalues and the Calabi flow
• Monotonicity of eigenvalues of geometric operaters along the Ricci-Bourguignon flow
• Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds
• Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds