Applications of some elliptic equations in Riemannian manifolds. (English) Zbl 1310.58012
Summary: Let \((M^{n+1},g)\) be a compact Riemannian manifold with smooth boundary B and nonnegative Bakry-Emery Ricci curvature. In this paper, we use the solvability of some elliptic equations to prove some estimates of the weighted mean curvature and some related rigidity theorems. As their applications, we obtain some lower bound estimate of the first nonzero eigenvalue of the drifting Laplacian acting on functions on B and some corresponding rigidity theorems.
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 53C24 | Rigidity results |
| 35R01 | PDEs on manifolds |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
Riemannian manifold; Bakry-Emery Ricci curvature; rigidity theorem; eigenvalue; weighted mean curvatureReferences:
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