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Positive solutions to Schrödinger equations and geometric applications

  • Ovidiu Munteanu , Felix Schulze ORCID logo EMAIL logo and Jiaping Wang


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.

Funding source: Leverhulme Trust

Award Identifier / Grant number: VP2-2018-029

Award Identifier / Grant number: RPG-2016-174

Award Identifier / Grant number: DMS-1506220

Funding statement: 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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Received: 2020-07-23
Revised: 2020-10-12
Published Online: 2020-12-15
Published in Print: 2021-05-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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