论文标题
长期估计啤酒歧管上的热流
Long-time estimates for heat flows on ALE manifolds
论文作者
论文摘要
我们考虑与作用于渐近局部欧几里得（ALE）歧管上的矢量束相关的热方程式。建立了新颖的$ l^p-l^Q $衰减估计值，使Schrödinger运营商具有非平凡的$ L^2 $ -KERNEL。在一般的几何环境中，我们还证明了任意顺序的空间衍生物的新衰减估计。我们的主要动机是应用于非线性几何方程的稳定性，主要是RICCI流，该流程将在同伴论文中呈现。本文中的论点是，许多几何施罗丁•运算符可以写为狄拉克型操作员的平方。通过王的显着结果，在平行旋转器的假设下，对于Lichnerowicz Laplacian来说，这甚至是正确的。我们的分析基于弗雷德·霍尔姆理论的新型组合，用于dirac类型运营商在非紧缩歧管上的热核研究中的最新进展。
We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schrödinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.
