讲座题目： A dual Yamabe flow and related integral flows

主办单位：三峡数学研究中心/理学院

报告专家：熊金钢  教授

报告时间：2024年9月26日（周四）下午：14:00

报告地点：腾讯会议 224 243 986

报告人简介：熊金钢，北京师范大学教授、博导。研究领域为非线性分析（变分法、椭圆与抛物型偏微分方程），解决Berryman和Holland 1980年提出的快速扩散方程正则性猜想；解决Monge-Ampère方程、Yamabe方程以及奇异调和映照中的若干长期未解的奇点结构问题；开启非局部Nirenberg问题等临界非线性问题的研究。迄今在国际期刊《Amer. J. Math.》, 《J. Eur. Math. Soc.》, 《J. Reine Angew. Math.》，《Proc. London Math. Soc.》等发表论文近 40 篇，先后获国家优秀青年科学基金和国家杰出青年科学基金资助。

报告摘要：We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a dual Q curvature we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.