Abstract
Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.
Highlights
In the past few decades, many interesting properties of eigenvalues of some self-adjoint elliptic operators such as the usual Laplace operator, the p-Laplace operator, the biharmonic operator and so on have been investigated in fixed Riemannian metrics
Motivated by the work of Perelman [4] and Cao [5], research on eigenvalues of the Laplace operator and some other deformations related to the Laplace operator such as p-Laplacian and WittenLaplacian under various geometric flows such as the Ricci flow, the mean curvature flow (MCF), the Yamabe flow and the Gaussian curvature flow has always been an active area in the study of geometry and topology of manifolds during these years
Zhao [6] considered a compact, strictly convex two-dimensional surface without boundary smoothly immersed in 3 and proved that the first eigenvalue of the Laplace operator is nonincreasing along the unnormalized powers of the MCF if the initial two-dimensional surface is totally umbilical
Summary
In the past few decades, many interesting properties of eigenvalues of some self-adjoint elliptic operators such as the usual Laplace operator ( called Laplace-Beltrami operator), the p-Laplace operator ( called p-Laplacian), the biharmonic operator and so on have been investigated in fixed Riemannian metrics (see [1,2,3]).
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