First eigenvalues of geometric operators under the Yamabe flow

TL;DR

This paper investigates how the first eigenvalues of geometric operators evolve under the Yamabe flow on compact Riemannian and CR manifolds, providing estimates and monotonicity results under certain curvature conditions.

Contribution

It establishes new bounds for the first eigenvalue of the Laplacian using the Yamabe flow and proves monotonicity of eigenvalues along the flow under curvature assumptions.

Findings

First eigenvalue estimates in terms of Yamabe metrics.

Monotonicity of eigenvalues along the Yamabe flow.

Extension of results to manifolds with boundary and CR manifolds.

Abstract

Suppose $(M,g_0)$ is a compact Riemannian manifold without boundary of dimension $n\geq 3$. Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of $g_0$ with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.