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The first eigenvalue of Laplace operator under powers of mean curvature flow

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Abstract

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along powers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in ℝn+1, we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k-flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.

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Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Liang Zhao

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  1. Liang Zhao
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Correspondence to Liang Zhao.

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Zhao, L. The first eigenvalue of Laplace operator under powers of mean curvature flow. Sci. China Math. 53, 1703–1710 (2010). https://doi.org/10.1007/s11425-010-3123-7

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  • DOI: https://doi.org/10.1007/s11425-010-3123-7

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