Eigenvalues of the drifted Laplacian on complete metric measure spaces

Abstract

I In this paper, first we study a complete smooth metric measure space $(M^n,g,e^{-f}dv)$ with the ($\infty$)-Bakry-Émery Ricci curvature ${\text{Ric}}_f\ge \frac{a}{2}g$ for some positive constant $a$. It is known that the spectrum of the drifted Laplacian ${\Delta }_f$ for $M$ is discrete and the first nonzero eigenvalue of ${\Delta }_f$ has lower bound $\frac{a}{2}$. We prove that if the lower bound $\frac{a}{2}$ is achieved with multiplicity $k\ge 1$, then $k\le n$, $M$ is isometric to ${\Sigma }^{n-k}\times {\mathbb{R}}^k$ for some complete $(n-k)$-dimensional manifold $\Sigma$ and by passing an isometry, $(M^n,g,e^{-f}dv)$ must split off a gradient shrinking Ricci soliton $({\mathbb{R}}^k,g_{can},\frac{a}{4}|t{|}^2)$, $t\in {\mathbb{R}}^k$. This result has an application to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian $\mathcal{L}$ for properly immersed self-shrinkers in the Euclidean space ${\mathbb{R}}^{n+p}$, $p\ge 1$ and show the discreteness of the spectrum of $\mathcal{L}$ and a logarithmic Sobolev inequality.

24 pages. References updated; new theorem (Theorem 3) added. Corresponding changes were made

References in corpus (7)

• Volume growth, eigenvalue and compactness for self-shrinkers
• Smooth metric measure spaces with non-negative curvature
• Eigenvalue estimate and compactness for closed $f$-minimal surfaces
• $\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow
• The essential spectrum of the Laplacian
• Geometry of manifolds with densities
• The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension

Cited by in corpus (1)

• Stability properties and gap theorem for complete $f$-minimal hypersurfaces