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Spectrum of the drift Laplacian on Ricci expanders

Helton Leal, Matheus Vieira, Detang Zhou

Abstract

In this paper, we study the spectrum of the drift Laplacian on Ricci expanders. We show that the spectrum is discrete when the potential function is proper, and we show that the hypothesis on the properness of the potential function cannot be removed. We also extend previous results concerning the asymptotic behavior of the potential function on Ricci expanders. This allows us to conclude that the drift Laplacian has discrete spectrum on Ricci expanders whose Ricci curvature is bounded below by a suitable constant, possibly negative. Further, we compute all the eigenvalues of the drift Laplacian on rigid expanders and rigid shrinkers. Lastly, we investigate the second eigenvalue of the drift Laplacian on rigid Ricci expanders whose Einstein factor is a closed hyperbolic Riemann surface.

Spectrum of the drift Laplacian on Ricci expanders

Abstract

In this paper, we study the spectrum of the drift Laplacian on Ricci expanders. We show that the spectrum is discrete when the potential function is proper, and we show that the hypothesis on the properness of the potential function cannot be removed. We also extend previous results concerning the asymptotic behavior of the potential function on Ricci expanders. This allows us to conclude that the drift Laplacian has discrete spectrum on Ricci expanders whose Ricci curvature is bounded below by a suitable constant, possibly negative. Further, we compute all the eigenvalues of the drift Laplacian on rigid expanders and rigid shrinkers. Lastly, we investigate the second eigenvalue of the drift Laplacian on rigid Ricci expanders whose Einstein factor is a closed hyperbolic Riemann surface.

Paper Structure

This paper contains 9 sections, 19 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth metric measure spaces
  4. Spectrum of the drift Laplacian
  5. Ricci solitons
  6. Discreteness of drift Laplacians on Ricci expanders
  7. The spectrum of the drift Laplacian on rigid Ricci expanders
  8. The second eigenvalue of the drift Laplacian on rigid Ricci expanders
  9. Appendix: Spectrum on product manifolds

Key Result

Theorem 1.1

Let $\left(M,g,f\right)$ be a complete Ricci expander. Suppose that the potential function $f$ is proper. Then: (i) The spectrum of the drift Laplacian $\Delta_{f}$ in the space $L_{f}^{2}$ is discrete. (ii) Assuming that the scalar curvature $S$ is bounded above by a constant, the spectrum of the d

Theorems & Definitions (30)

  • Theorem 1.1: Theorem \ref{['thm:discrete']}
  • Theorem 1.2: Theorem \ref{['thm:asymptotic']}
  • Corollary 1.3: Corollary \ref{['cor:ricci']}
  • Theorem 1.4: Theorem \ref{['thm:rigid']}
  • Theorem 1.5: Theorem \ref{['thm:second']}
  • Remark 1.6
  • Theorem 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • ...and 20 more