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Estimate and monotonicity of the first eigenvalue under the Ricci flow

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Abstract

In this paper, we first derive a monotonicity formula for the first eigenvalue of \({-\Delta +aR (0 < a \leq 1/2)}\) on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    **aodong Cao

  2. Department of Applied Mathematics, College of Science, East Campus, China Agricultural University, Bei**g, 100083, People’s Republic of China

    Songbo Hou

  3. Department of Mathematics, Utah Valley University, Orem, UT, 84058, USA

    Jun Ling

Authors
  1. **aodong Cao
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  2. Songbo Hou
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  3. Jun Ling
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Correspondence to **aodong Cao.

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Cao, X., Hou, S. & Ling, J. Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451–463 (2012). https://doi.org/10.1007/s00208-011-0740-6

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  • DOI: https://doi.org/10.1007/s00208-011-0740-6

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