Paper · Jun 24, 2020
Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons
As on arXiv
Abstract
On gradient shrinking Ricci solitons, we observe that the study of Schrödinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green's function of the Schrödinger operator. We also prove sharp lower bounds for eigenvalues of the Schrödinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.
- Submitted
- Jun 24, 2020
- Last revised
- Sep 26, 2022 (v2)
- Subjects
- math.DG
Comments: revised version, accepted by Manuscripta Math
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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.3The Heat Equation on ℝⁿ
The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.
- 6A.4Fourier Methods and Scale Space
Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.