Positive solutions to Schrodinger equations and geometric applications

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrodinger 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.

Automated summary

A variant of Li-Tam theory is developed in this study, linking each end of a complete Riemannian manifold to a positive solution of a specific Schrodinger equation on the manifold. It is shown that these positive solutions must exhibit polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. The finiteness of the number of ends is then derived as a consequence. Additionally, a direct proof of the finiteness result is provided in the case of a specific type of Sobolev inequality, and an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained as an application.