Variational status of a class of fully nonlinear curvature prescription problems

双曲率预设问题的平均场问题方法

A mean field problem approach for the double curvature prescription problem

本文提出一种新的平均场型公式研究带边界紧致曲面的双曲率预设问题，将其转化为带非线性Neumann条件的Liouville型PDE，并通过变分技术得到欧拉示性数为正、零、负三种情形下的存在性结果。

The weighted $σ_k$-curvature of a smooth metric measure space

We propose a definition of the weighted $σ_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $σ_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted $σ_k$-curvature functional. We also discuss related conjectures for weighted Einstein manifolds.

规定测地线与Riemann度量的变分问题

Prescribing geodesics and a variational problem for Riemannian metrics

研究给定流形上无参数路径是否与Riemann度量的测地线一致，提出非负泛函并分析其变分方程，揭示其与Yamabe类型的联系，证明在曲面情况下每个共形类中存在唯一的共形临界度量。