Talk I

Title: Sharp estimates of the heat kernel and Green’s function on the complete manifold with nonnegative Ricci curvature

Abstract: In this talk, we discuss the global behaviors of the heat kernel and Green's function one the complete manifold with nonnegative Ricci curvature.  we first obtain sharp two-side Gaussian bounds for the heat kernel that sharpens the well-known Li-Yau’s two-side bounds, based on the sharp Li-Yau’s Harnack inequality on such a manifold. As an application, we get the optimal gradient and Laplacian estimates on the heat kernel with our new Hamilton’s estimates of positive bounded solutions. Next, when the manifold has Euclidean volume growth, we obtain the new pointwise lower and upper bounds for the heat kernel in term of a natural geometric quantity that characterizes the decay rate of the normalized Bishop–Gromov quantities. And as applications of the two side bounds, we obtain the large-time asymptotics of the heat kernel, which extends the results of P. Li and Li-Tam-Wang, and the large-scale behavior of Green’s function, which extends a result of Colding-Minicozzi, on the complete manifold with nonnegative Ricci curvature and Euclidean volume growth.

Talk II

Title: Heat kernel Gaussian bounds, Hamilton's estimates and monotonicity of entropy formulas on the complete manifolds with negative curvature

Abstract: In this talk, we first discuss sharp two-side Gaussian bounds for the heat kernel on the complete manifolds with negative curvature lower bound, which sharpens the well-known two-side heat kernel bounds of Li-Yau and Strum. Next, we obtain Hamilton's estimates on the gradient and Laplacian for the bounded positive solution of the heat equation on the complete manifold, which extends the Hamilton's results on the compact manifold, and the sharp gradient and Laplacian estimates for the heat kernel as applications. The last part of talk, we discuss various monotonicity of entropy  formulas for the positive solution of the heat equation and the Perelman type differential Harnack inequalities for the heat kernel on the complete manifolds with negative curvature lower bound, which extends the results of Ni on the compact manifold with nonnegative Ricci curvature.

Talk III

Title: Characterization of Carleson measures via spectral estimates on compact manifolds with boundary

Abstract: In this talk, we first discuss the asymptotics of the reproducing kernel of the space $E_L$  and a Bernstein type inequality for $f\in E_L$, where $E_L\subset L^2(M)$ is generated by eigenfunctions of eigenvalues $<L$ associated to Dirichlet Laplacian and Neumann Laplacian on a compact Riemannian manifold $M$.  Next, under suitable convexity assumptions on the boundary, the mean value inequalities of subharmonic functions associated to $E_L$ in the scale $\frac{1}{\sqrt{L}}$ are achieved on the metric ball with possible nonempty intersection with the boundary, which generalizes the classical mean value inequality on the interior geodesic ball by Li, Schoen and Yau. Applying the asymptotic estimates, Bernstein type inequality and mean value inequality on these spaces $E_L$, we show a characterization of the $L^2$-Carleson measures associated to Neumann Laplacian, and give a counterexample to invalid the characterization of the $L^2$-Carleson measures.

报告人：徐向锦，美国Binghamton University副教授，研究方向是流形上的调和分析和非线性偏微分方程。