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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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