Paper · Jul 14, 2020

Positive solutions to Schrödinger equations and geometric applications

Ovidiu Munteanu, Felix Schulze, Jiaping Wang

math.DGarXiv:2007.07191v3
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Abstract

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

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Submitted
Jul 14, 2020
Last revised
Nov 10, 2020 (v3)
Subjects
math.DG, math.AP

Comments: Final version, to appear in Crelle. 31 pages

Ricci solitons

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)1Munteanu 2016 Poisson equation on complete manifolds (Ovidiu Munteanu, Chiung-Jue Anna Sung and Jiaping Wang, 2017)2Munteanu 2017 Generic uniqueness of expanders with vanishing relative entropy (Alix Deruelle and Felix Schulze, 2018)8Deruelle 2018 Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons (Jia-Yong Wu and Peng Wu, 2019)4Wu 2019 Geometric inequalities and rigidity of gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)7Wu 2020 A relative entropy and a unique continuation result for Ricci expanders (Alix Deruelle and Felix Schulze, 2021)5Deruelle 2021 A Minkowski type inequality for manifolds with positive spectrum (Ovidiu Munteanu and Jiaping Wang, 2023)3Munteanu 2023 Uniqueness of Ricci flows from singular initial data (Laura Flower, Man-Chun Lee and Felix Schulze, 2026)6Flower 2026
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  1. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

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  4. Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

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  6. Uniqueness of Ricci flows from singular initial data

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  7. Geometric inequalities and rigidity of gradient shrinking Ricci solitons

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  8. Generic uniqueness of expanders with vanishing relative entropy

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.