Paper · Aug 18, 2022
Perelman's functionals on manifolds with non-isolated conical singularities
As on arXiv
Abstract
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 -functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator 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 -functional on these singular manifolds, and further, to prove that the infimum of -functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the -functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of and Perelman's functionals on manifolds with more general singularities, i.e. the -horn singularities which serve as prototypes of algebraic singularities.
- Submitted
- Aug 18, 2022
- Last revised
- Nov 12, 2023 (v2)
- Subjects
- math.DG
Comments: Sections are re-ordered, some proofs are simplified, some typos and errors are corrected, revised version, 58 pages
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6B.16Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.