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First eigenvalues of geometric operators under the Yamabe flow

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Abstract

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.

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Authors and Affiliations

  1. Department of Mathematics, Sogang University, Seoul, 121-742, Korea

    Pak Tung Ho

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  1. Pak Tung Ho
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Correspondence to Pak Tung Ho.

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This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 201531021.01).

This is the shortened version of the paper. The full and detailed version is posted on arXiv.org [28].

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Ho, P.T. First eigenvalues of geometric operators under the Yamabe flow. Ann Glob Anal Geom 54, 449–472 (2018). https://doi.org/10.1007/s10455-018-9608-2

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