The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3657–3689 | Cite as

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

  • Xianzhe DaiEmail author
  • Changliang Wang


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.


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

Mathematics Subject Classification

53C44 58J50 



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


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Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiChina
  2. 2.University of CalifornaiSanta BarbaraUSA
  3. 3.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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