# Eigenvalue Estimates for Beltrami-Laplacian Under Bakry-Émery Ricci Curvature Condition

@article{Wu2021EigenvalueEF, title={Eigenvalue Estimates for Beltrami-Laplacian Under Bakry-{\'E}mery Ricci Curvature Condition}, author={Ling Wu and Xin Song and Meng Zhu}, journal={Potential Analysis}, year={2021} }

On closed Riemannian manifolds with Bakry-Émery Ricci curvature bounded from below and bounded gradient of the potential function, we obtain lower bounds for all positive eigenvalues of the Beltrami-Laplacian instead of the drifted Laplacian. The lower bound of the kth eigenvalue depends on k, Bakry-Émery Ricci curvature lower bound, the gradient bound of the potential function, and the dimension and diameter upper bound of the manifold, but the volume of the manifold is not involved…

## References

SHOWING 1-10 OF 29 REFERENCES

### Eigenvalue Estimates on Bakry–Émery Manifolds

- Mathematics
- 2015

We demonstrate lower bounds for the eigenvalues of compact Bakry–Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalized maximum principle which…

### Gradient and Eigenvalue Estimates on the Canonical Bundle of Kähler Manifolds

- MathematicsThe Journal of Geometric Analysis
- 2021

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kähler manifolds, where m…

### Rigidity of manifolds with Bakry–Émery Ricci curvature bounded below

- Mathematics
- 2012

Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator $${\Delta_f =…

### Convergence of Ricci flows with bounded scalar curvature

- Mathematics
- 2016

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature…

### Heat kernel on smooth metric measure spaces and applications

- Mathematics
- 2014

We derive a Harnack inequality for positive solutions of the f-heat equation and Gaussian upper and lower bound estimates for the f-heat kernel on complete smooth metric measure spaces with…

### Eigenvalue Comparison on Bakry-Emery Manifolds

- Mathematics
- 2011

We prove a comparison theorem on the modulus of continuity of the solution of a heat equation with a drifting term on Bakry-Emery manifolds. A direct consequence of the result is an alternate proof…

### Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry–Émery Ricci Curvature II

- Mathematics
- 2013

Let $${L=\Delta-\nabla\varphi\cdot\nabla}$$ be a symmetric diffusion operator with an invariant measure $${d\mu=e^{-\varphi}dx}$$ on a complete Riemannian manifold. In this paper we prove Li–Yau…

### Bounds on Harmonic Radius and Limits of Manifolds with Bounded Bakry–Émery Ricci Curvature

- MathematicsThe Journal of Geometric Analysis
- 2018

AbstractUnder the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $$C^{\alpha } \cap…

### Regularity of Kähler–Ricci flows on Fano manifolds

- Mathematics
- 2013

In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano n-manifolds with Ricci curvature bounded in Lp-norm for some $${p > n}$$p>n. Using this regularity theory, we…

### Analysis of weighted Laplacian and applications to Ricci solitons

- Mathematics
- 2011

We study both function theoretic and spectral properties of the weighted Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-\'{E}mery curvature $Ric_f$…