# MONOTONICITY OF THE FIRST EIGENVALUE OF THE LAPLACE AND THE p-LAPLACE OPERATORS UNDER A FORCED MEAN CURVATURE FLOW

• Mao, Jing
• Accepted : 2018.02.21
• Published : 2018.11.01
• 54 14

#### Abstract

In this paper, we would like to give an answer to Problem 1 below issued firstly in [17]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

#### Keywords

Ricci-Hamilton flow;mean curvature flow;Laplace operator;p-Laplace operator

#### Acknowledgement

Supported by : NSF of China, Fok Ying-Tong Education Foundation