Monotonicity formulas for the first eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow

Abstract

Let p,φ be the weighted p-Laplacian defined on a smooth metric measure space. We study the evolution and monotonicity formulas for the first eigenvalue, λ 1 = λ(p,φ), of p,φ under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that λ 1 is monotonically nondecreasing and almost everywhere differentiable along the flow existence.

On the spectrum of the weighted p-Laplacian under the Ricci-harmonic flow

“…Throughout, we will consider an n-dimensional complete Riemannian manifold (M, g, dμ) equipped with weighted measure dμ = e -φ dv and potential function φ ∈ C ∞ (M, dμ), whose metric g = g(t) evolves along either the Ricci-harmonic flow or volume-preserving Ricci-harmonic flow. Firstly, we extend results in [8] to the case of volume-preserving Ricci-harmonic flow. We will obtain a variation formula for the first eigenvalue and show that it is monotonically increasing under this setup.…”

“…Cao [11] and Li [18] extended Perelman's result with or without any curvature assumption. Recently, [8] was motivated by the first author's papers [5] and [1] where he studied the evolution and monotonicity of the first eigenvalue of the p-Laplacian and weighted Laplacian, respectively. In [14], Di Cerbo proved that the first eigenvalue of Laplace-Beltrami operator on a 3-dimensional closed manifold with positive Ricci curvature diverges as t → T under the Ricci flow.…”

“…Motivated by [14] and [5], we will show the same result for the eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow and on gradient almost Ricci-harmonic soliton for p = 2. Meanwhile, we will first extend the results of [8] to the case of a volume-preserving flow.…”

“…The Riemannian metric g(x) at any point x ∈ M is a bilinear symmetric positive definite matrix. As in [8], we denote a symmetric 2-tensor by Sc := Rc -α∇φ ⊗ ∇φ and its metric trace by S := R -α|∇φ| 2 , where R is the scalar curvature of (M, g) and ∇ i φ = ∂ ∂x i φ. We denote the Laplace-Beltrami operator on (M, g) by .…”

“…When p = 2, this is just the Witten Laplacian, and when φ is a constant, it is just the p-Laplacian. See [8] for detailed descriptions of Witten Laplacian and p-Laplacian.…”