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Positive solutions to Schrödinger equations and geometric applications

Munteanu, O; Schulze, F; Wang, J; (2020) Positive solutions to Schrödinger equations and geometric applications. Journal fur die Reine und Angewandte Mathematik 10.1515/crelle-2020-0046. (In press).

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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.

Type: Article
Title: Positive solutions to Schrödinger equations and geometric applications
DOI: 10.1515/crelle-2020-0046
Publisher version: https://doi.org/10.1515/crelle-2020-0046
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10109642
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