Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces
Shouwen Fang , Liang Zhao , Peng Zhu
Communications in Mathematics and Statistics ›› 2016, Vol. 4 ›› Issue (2) : 217 -228.
Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces
In the paper we first derive the evolution equation for eigenvalues of geometric operator $-\Delta _{\phi }+cR$ under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M, where $\Delta _{\phi }$ is the Witten–Laplacian operator, $\phi \in C^{\infty }(M)$, and R is the scalar curvature. We then prove that the first eigenvalue of the geometric operator is nondecreasing along the Ricci flow on closed surfaces with certain curvature conditions when $0<c\le \frac{1}{2}$. As an application, we obtain some monotonicity formulae and estimates for the first eigenvalue on closed surfaces.
First eigenvalue / Witten–Laplacian / Ricci flow
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