Lectures on Capacities and Dispersions within Manifolds

Professor Jie Xiao (Memorial University）

2025.11.4-2025.11.13

摘要：Five lectures will concentrate on recent 5 papers (whose 4 are joint with Xiaoshang Jin) on the capacities, masses, areas, volumes and dispersions living within complete Riemanniann manifolds.

专家简介: 1992年博士毕业于北京大学数学科学学院，现为加拿大纽芬兰纪念大学教授，博士生导师(University Research Professor), 科学院院长杰出学者奖获得者, 2025年当选加拿大数学会院士(CMS Fellow), 国际著名数学家。主要研究兴趣为调和分析、几何分析、偏微分方程、函数空间及算子理论。特别地，肖杰教授致力于调和分析，复分析和算子理论及其交叉应用，但也将函数空间和位势理论的方法和技术应用于共形微分几何、热方程、纳维-斯托克斯方程和波动方程的研究。迄今为止，肖杰教授有200多篇文章发表在国际领先的数学期刊上，包括《Adv. Math.》、《J. Math. Pures Appl.》、《J. Eur. Math. Soc》等。 同时，肖杰教授已发表6本教材和专著。此外，肖杰教授还担任《Advances in Analysis and Geometry》和《Canadian Mathematical Bulletin》期刊的主编，以及《Journal of Mathematical Analysis and Applications》的部门编辑。

第1讲: Harmonic capacities of asymptotically flat three-manifolds

2025年11月4日周二8：30-12：00, 数学楼4106

第1讲摘要：This lecture demonstrates an optimal relationship between $(1,3)-\ni p$-harmonic capacity and the surface area of an asymptotically flat $3$-manifold with nonnegative scalar curvature, and its consequence involving the total mass of manifold and the $p$-harmonic capacity in terms of Willmore's energy.

第2讲: Heat dispersion laws in smooth compact manifolds

2025年11月6日周四14：00-17：30, 数学楼4108

第2讲摘要：Given a Lipschitz conductor $K$ in the smooth compact Riemannian $2\le n$-manifold $(M,g)$, such a half generic heat dispersion law $H^d_{p,\Phi,\Psi}(K,M)=2^{-1}H^d_{\Delta_p,\Phi,\Psi}(K,M)$ is not only newly-established but also deeply-explored via not only a comparison law for the generic heat dispersion but also a recycling law for the quasilinear Laplace-Robin eigenvalue.

第3讲: Essential capacity-volume estimates for rotationally symmetric manifolds

2025年11月7日周五14：00-17：30, 数学楼4107

第3讲摘要：Given $p\in [1,\infty]$, this lecture presents the novel basic volumetric estimates for the relative $p$-capacities with their applications to finding not only the sharp weak $(p,q)$-imbeddings but also the precise lower bounds of the principal $p$-frequencies, which principally live in the rotationally symmetric manifolds.

第4讲: Capacities-to-masses for Laplace-Beltrami operators on complete Riemannian manifolds

2025年11月11日周二8：30-12：00, 数学楼4106

第4讲摘要：This lecture presents an innovative approach to the capacities-masses for the Laplace-Beltrami operators on the complete Riemannian manifolds, unexpectedly solving an open problem on the limiting variational capacity.

第5讲: Sharply estimating hyperbolic capacities

2025年11月13日周四14：00-17：30, 数学楼4108

第5讲摘要：This lecture is devoted to establish four types of sharp capacitary inequalities within the hyperbolic space.