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.

中文翻译：

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Schrödinger方程和几何应用的正解

开发了一种Li–Tam理论的变体，该变体与一个完整的黎曼流形的每一端相关，该流形上给定Schrödinger方程的一个正解。证明了这样的正解必须是在适当的尺度不变Sobolev不等式下的固定阶多项式增长。因此，得出了端数的有限性结果。如果Sobolev不等式是特定类型，则直接证明有限性结果。作为一种应用，获得了对递减的梯度Ricci孤子和欧几里得空间的子流形的末端数量的估计。