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.
Funding source: Leverhulme Trust
Award Identifier / Grant number: VP2-2018-029
Award Identifier / Grant number: RPG-2016-174
Funding source: National Science Foundation
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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