Perelman’s \(\lambda \)-Functional on Manifolds with Conical Singularities

Abstract

In this paper, we prove that on a compact manifold with isolated conical singularity, the spectrum of the Schrödinger operator \(-4\Delta +R\) consists of discrete eigenvalues with finite multiplicities, if the scalar curvature R satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectral properties, we extend the theory of the Perelman’s \(\lambda \)-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.

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Funding

Funding were provided by Directorate for Mathematical and Physical Sciences (Grant No. 1611915) and Simons Foundation.

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Correspondence to Xianzhe Dai.

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Dai, X., Wang, C. Perelman’s \(\lambda \)-Functional on Manifolds with Conical Singularities. J Geom Anal 28, 3657–3689 (2018). https://doi.org/10.1007/s12220-017-9971-4

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Keywords

  • Perelman’s F entropy
  • Perelman’s Lambda functional
  • Manifolds with conical singularity
  • Eigenvalues and eigenfunctions

Mathematics Subject Classification

  • 53C44
  • 58J50