GAP THEOREMS FOR ENDS OF SMOOTH METRIC MEASURE SPACES

Article Mathematics, Applied

THE SPLITTING THEOREM AND TOPOLOGY OF NONCOMPACT SPACES WITH NONNEGATIVE N-BAKRY' EMERY RICCI CURVATURE

Alice Lim

Summary: This paper generalizes topological results for spaces with nonnegative N-Bakry' Emery Ricci curvature, extending known results for noncompact manifolds with nonnegative Ricci curvature. The study focuses on the Splitting Theorem and the geodesic loops to infinity property. Additionally, it is shown that the relevant cohomology group is 0 for complete, noncompact Riemannian manifolds with nonnegative N-Bakry' Emery Ricci curvature.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2021)

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Article Mathematics

Positive solutions to Schrodinger equations and geometric applications

Ovidiu Munteanu et al.

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.

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK (2021)

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Article Mathematics, Applied

SPLITTING THEOREM FOR RICCI SOLITON

Guoqiang Wu

Summary: For a gradient Ricci soliton (M, g, f) with certain conditions on a geodesic line gamma, the manifold will split off a line isometrically.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2021)

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Article Mathematics