https://neckpinch.com/authors/changliang-wang/New arXiv papers by Changliang Wang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2208.08776/arxiv:2208.08776Thu, 18 Aug 2022 00:00:00 GMTXianzhe Dai, Changliang Wang — In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's λ-functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator −4Δ+ R on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the λ-functional on these singular manifolds, and further, to prove that the infimum of W-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the W-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of −4Δ+ R and Perelman's functionals on manifolds with more general singularities, i.e. the r^α-horn singularities which serve as prototypes of algebraic singularities.SingularitiesPerelman entropyScalar curvatureEigenvalues & spectrum