Harnack's Inequality in Riemannian Geometry

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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.