Convergence of Riemannian manifolds and Laplace operators

黎曼流形被认为是配备黎曼距离的度量空间。从这个角度来看，一组紧致的，连通的黎曼流形具有由Gromov-Hausdorff距离定义的一致结构，并且围绕黎曼流形的收敛理论有大量的活动，其中包括从谱几何和扩散过程的角度进行的一些工作。1994年，我们在一组紧致的（加权）黎曼流形上引入了谱距离，用热核代替黎曼距离，并证明了黎曼流形谱收敛的一些结果。在本项目中，我们继续研究并证明了以下结果：（1）能量形式在一定意义上“支配”了内禀距离，支配关系可以用极限空间的能量密度、体积加倍性质和尺度不变的Poincare不等式来表示，它们在我们的理论中起着重要的作用。(2)不仅函数空间上的能量泛函，而且映射空间上的能量泛函都可以用同样的方法来讨论，并且可以根据所研究的空间的几何和拓扑性质来研究泛函的收敛性。(3)黎曼向量丛及其上的能量泛函自然成为我们理论的重要课题。(4)变形的子流形提供了新的问题，在我们的设置。

Riemannian manifolds are considered as metric spaces equipped with Riemannian distances. From this point of view, a set of compact, connected Riemannian manifolds has uniform structure defined by the Gromov-Hausdorff distance, and there are intensive activities around the convergence theory of Riemannian manifolds, which include some works from the viewpoint of spectral geometry and also diffusion processes. In 1994, we introduced a spectral distance on a set of compact, (weighted) Riemannian manifolds, using heat kernels instead of Riemannian distances, and proved some results on the spectral convergence of Riemannian manifolds. In this project, we I continued the study for further developments and proved some results as follows: (1) the energy forms "dominates" the intrinsic distances in a certain sense; the relation of domination can be expressed in terms of the energy density in the limit spaces, the volume doubling property, and the scale invariant Poincare inequality, which play important roles in our theory. (2) the energy functional not only on function spaces but also on the space of maps are able to be discussed in the same vein and the convergence of the functional can be investigated in relation to the geometric and topological properties of spaces under study. (3) Riemannian vector bundles and the energy functional on them can naturally arise as the important subject of our theory. (4) Deformations of submanifolds provide new problems in our setting.