Eigenvalues of the drifted Laplacian on complete metric measure spaces
By Xu Cheng, Detang Zhou. 2013
71 citations recorded by the source corpus.
I In this paper, first we study a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ with the ($\infty$)-Bakry-\'Emery Ricci curvature $\textrm{Ric}_f\ge \frac a2g$ 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 $\frac a2$. We prove that if the lower bound $\frac a2$ is achieved with multiplicity $k\geq 1$, then $k\leq n$, $M$ 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 $\mathbb{R}^{n+p}$, $p\geq1$ and show the discreteness of the spectrum of $\mathcal{L}$ and a logarithmic Sobolev inequality.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7fb24fa72391f0d70f6f2655804f1c9635d3699a
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Complexity of parabolic systems (2019, methodology): “ COMPLEXITY OF PARABOLIC SYSTEMS We next estimate the drift heat kernel H on a shrinker in arbitrary codimension. To construct H, we need that the spectrum is discrete; this was proven by Cheng-Zhou, [CxZh]. Theorem 5.3. Let Σn ⊂ RN be a shrinker with finite entropy. There is a complete basis of W1,2”