[Paper Review] First eigenvalue of the $p$-Laplace operator along the Ricci flow

Jia-Yong Wu, Ermin Wang+1 more|arXiv (Cornell University)|Dec 24, 2009

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper investigates the evolution of the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed Riemannian manifolds. Under curvature assumptions, it proves the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere. For orientable closed surfaces with negative Euler characteristic, it establishes differentiability without curvature assumptions and derives a $p$-eigenvalue comparison theorem.

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.

Motivation & Objective

• To analyze the continuity, monotonicity, and differentiability of the first eigenvalue of the $p$-Laplace operator under the Ricci flow on closed manifolds.
• To establish conditions under which the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere.
• To derive a $p$-eigenvalue comparison-type theorem for orientable closed surfaces with negative Euler characteristic, without curvature assumptions.
• To generalize known results for the Laplacian ($p=2$) to the nonlinear $p$-Laplacian under geometric flows.
• To construct monotonic quantities along the Ricci and Yamabe flows to analyze eigenvalue evolution.

Proposed method

• Utilizes the unnormalized and normalized Ricci flow equations to study the evolution of the $p$-Laplacian eigenvalue.
• Applies variational characterization of the first $p$-eigenvalue via the Rayleigh quotient for the $p$-Laplace operator.
• Derives a differential formula for the first $p$-eigenvalue along the Ricci flow using the eigenfunction and metric evolution.
• Employs maximum principle techniques to bound scalar curvature and control the growth of eigenvalues.
• Constructs monotonic quantities involving $\lambda_{1,p}(t)$, scalar curvature bounds $\rho_0$, $\sigma_0$, and time-dependent scaling factors.
• Adapts methods from prior works on $p=2$ to the nonlinear $p$-Laplacian case, extending results to $p \neq 2$.

Experimental results

Research questions

• RQ1 Under what curvature conditions is the first $p$-eigenvalue of the $p$-Laplace operator strictly increasing along the Ricci flow?
• RQ2 Is the first $p$-eigenvalue differentiable almost everywhere along the Ricci flow, and under what geometric assumptions?
• RQ3 Can a $p$-eigenvalue comparison theorem be established for closed surfaces with negative Euler characteristic without curvature assumptions?
• RQ4 How do monotonic quantities involving $\lambda_{1,p}(t)$ and scalar curvature bounds evolve under the Ricci and Yamabe flows?
• RQ5 What is the precise differential formula for the first $p$-eigenvalue along the Ricci flow, and how does it relate to the metric and curvature?

Key findings

• The first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under suitable curvature assumptions.
• For orientable closed surfaces with negative Euler characteristic, the first $p$-eigenvalue is differentiable almost everywhere without any curvature assumption.
• A $p$-eigenvalue comparison-type theorem is established for closed surfaces with negative Euler characteristic based on differentiability and monotonicity.
• Monotonic quantities of the form $\lambda_{1,p}(t) \cdot (1 - \rho_0 t)^{n/2} \cdot (1 - \sigma_0 t)^{(p-n)/2}$ are increasing for $1 < p < n$ along the unnormalized Yamabe flow.
• For $p \geq n$, the quantity $\lambda_{1,p}(t) \cdot (1 - \rho_0 t)^{p/2}$ is increasing along the unnormalized Yamabe flow.
• The quantities $\lambda_{1,p}(t) \cdot (1 - \sigma_0 t)^{p/2}$ and $\lambda_{1,p}(t) \cdot (1 - \sigma_0 t)^{(p-n)/2} \cdot (1 - \rho_0 t)^{n/2}$ are decreasing along the unnormalized Yamabe flow for $p \geq n$ and $1 < p < n$, respectively.