The First Eigenvalue of Witten-Laplacian on Manifolds with Time-Dependent Metrics

The main purpose of this paper is to discuss the monotonicity and differentiability for the first eigenvalue of Witten-Laplacian on closed metric measure spaces with time-dependent metrics. We show that the first eigenvalue of Witten-Laplacian is monotonous and differentiable almost everywhere along Ricci flow or modified Ricci flow with different potentials. We conclude some properties on gradient Ricci solitons and Ricci breathers as applications. Meanwhile, some monotonic quantities about the first eigenvalue of Witten-Laplacian are observed. Lastly, we derive a Witten-eigenvalue comparison theorem on closed surfaces. These conclusions are natural extension of some known results for Laplace–Beltrami operator under the Ricci flow.