The Loewner-Nirenberg Problem in Cones

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:2024-06-04 10:00





Loewner and Nirenberg discussed complete metrics conformal to the Euclidean metric and with a constant scalar curvature in bounded domains in the Euclidean space. The conformal factors blow up on boundary. The asymptotic behaviors of the conformal factors near boundary are known in C^2-domains. In this talk, we discussasymptotic behaviors near vertices of cones. We will prove that solutions on finite cones are well-approximated by the solution in the corresponding infinite cone. To derive optimal estimates, we need to study a class of elliptic operators over spherical domains. These operators are singular on boundary. We will study the eigenvalue problem with the homogeneous Dirichlet boundary value and investigate boundary behaviors of the eigenfunctions.  


韩青,美国圣母大学数学系终身教授。美国纽约大学库朗数学研究所博士,美国芝加哥大学博士后。获美国Sloan Research Fellowship。韩青教授长期致力于非线性偏微分方程和几何分析的研究,在等距嵌入、Monge-Ampere方程、调和函数的零点集和奇异集、退化方程等方面做出了一系列原创性的重要研究成果。