First eigenvalue monotonicity for the \(p\)-Laplace operator under the Ricci flow. (English) Zbl 1306.58003
In this paper, the author discusses the monotonicity of the first eigenvalue of the \(p\)-Laplace operator (\(p=2\)) along the Ricci flow on closed Riemannian manifolds. The main result is to prove that the first eigenvalue of the \(p\)-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions. Otherwise, L. Ma [Ann. Global Anal. Geom. 29, No. 3, 287–292 (2006; Zbl 1099.53046)] considered the eigenvalues of the Laplace operator along the Ricci flow. He discussed the monotonicity of the eigenvalues of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. He showed that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. In this paper, the author also extends some parts of Ma’s results.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Citations:
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