论文标题
在某些riemannian歧管上热半群的渐近行为
Asymptotic behavior of the heat semigroup on certain Riemannian manifolds
论文作者
论文摘要
我们表明，在非阴性RICCI曲率的完整，连接和非紧密的Riemannian歧管上，使用$ l^{1} $初始数据的热方程式解决方案，而初始数据则渐近地表现为热量元素的质量。与先前已知的结果相反，在负弯曲的环境中，不需要对初始数据的径向假设。类似的长期收敛结果对满足热核的Li-yau双面估计值的更通用歧管仍然有效。此外，我们提供了一个反例，使得这种渐近现象在具有两个欧几里得末端的歧管上失败了。
We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to the previously known results in negatively curved contexts, the radiality assumption on the initial data is not required. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. Moreover, we provide a counterexample such that this asymptotic phenomenon fails in sup norm on manifolds with two Euclidean ends.
