摘要: In the series of talks, on a geodesic ball with non-negative Ricci curvature and almost maximal volume, we construct the almost linear harmonic map on a concentric geodesic ball with uniformly small radius. We use the distance estimate formula on the geodesic ball and induction method to produce the harmonic map. And this intrinsic method gives the integral gradient estimate of harmonic map in term of the volume explicitly. As one application, we establish the quantitative difference estimate of two related harmonic functions defined on geodesic ball and Euclidean ball respectively. The talk will require one semester Riemannian geometry and one semester PDE course background. The details will be given. This is a joint work with Jie Zhou in Math Department of Tsinghua.