Harnack's Inequality in Riemannian Geometry

黎曼流形被认为是具有黎曼距离的度量空间，也是具有黎曼测度和能量形式的狄利克雷空间；在一类这样的流形上，Gromov-Hausdorff距离和谱距离诱导出一致拓扑。前者与度规结构有关，后者与谱结构有关。本文研究了一类紧连黎曼流形族，该类流形的热核满足一致对角估计，并从Gromov-Hausdorff距离和谱距离的角度，建立了该类极限空间结构的一些基本结果；(2)能量形式的收敛；(3)向量束拉普拉斯算子的谱收敛性；(4)族流形调和映射到非正曲率黎曼流形的收敛性。体积倍增条件和Neumann-Poincare不等式，即抛物型的Harnack不等式，在这些研究中发挥了重要作用。

Riemannian manifolds are considered as metric spaces equipped with Riemannian distances and also Dirichlet spaces endowed with the Riemannian measures and the energy forms ; on a family of such manifolds, the Gromov-Hausdorff distance and the spectral distance induce uniform topologies. The former is concerning the metric structure and the latter is concerning the spectral structure.In this program, we studied a family of compact, connected Riemannian manifolds such that the heat kernels of manifolds in the family satisfy a uniform on-diagonal estimate from below, and we have established some basic results on (1) the structure of limit spaces of the family from the view-points of Gromov-Hausdorff distance and the spectral distance ; (2) the convergence of energy forms ; (3) the spectral convergence of vector bundle Laplacians ; (4) the convergence of harmonic maps of manifolds in the family to Riemannian manifolds of nonpositive curvature. The volume doubling condition and the Neumann-Poincare inequality, equivalently the parabolic Harnack inequality, play important roles in these investigations.