论文标题
Schrödinger方程和几何应用的积极解决方案
Positive solutions to Schrödinger equations and geometric applications
论文作者
论文摘要
Li-Tam理论的一种变体，它与完整的Riemannian歧管的每一端相关联。证明这种积极的解决方案必须是在适当的缩放缩放不平等不等式下固定顺序的多项式生长。因此，随后是终点的有限结果。在Sobolev不平等的情况下，有限结果可以直接证明。作为一种应用，获得了缩小梯度RICCI孤子和欧几里得空间的亚构福利的末端数量的估计。
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.
