Positive solutions to Schrödinger equations and geometric applications

Ovidiu Munteanu, Felix Schulze, Jiaping Wang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)185-217
Number of pages33
JournalJournal fur die Reine und Angewandte Mathematik
Volume2021
Issue number774
DOIs
StatePublished - May 1 2021

Bibliographical note

Publisher Copyright:
© 2020 Walter de Gruyter GmbH, Berlin/Boston 2021 Leverhulme Trust VP2-2018-029 RPG-2016-174 National Science Foundation DMS-1506220 The first author was partially supported by NSF grant DMS-1506220 and by a Leverhulme Trust Visiting Professorship VP2-2018-029. The second author was supported by a Leverhulme Trust Research Project Grant RPG-2016-174.

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