几何中的退化与奇异偏微分方程

Since 1970s, a series of important geometric problems were solved through studies of nonlinear elliptic equations, including the Yamabe problem, the positive mass conjecture and the Calabi conjecture. With the perfection of the (strict) elliptic equations, degenerate and singular elliptic equations related to geometry and physics have received considerable attention and attracted many mathematicians. In the present project, we will study boundary blowup and boundary degeneracy problems involving Monge-Ampère equations and minimal surface equations, as well as asymptotic behaviors of solutions of the wave equation in the Schwarzschild space; we will study the existence of the extremal Kähler metrics in toric varieties and homogeneous toric bundles and study the solvability of the fourth order Abreu equation in polytopes; we will study the classification of solutions of degenerate Monge-Ampère equations and special Lagrangian equations in the half-space; we will study the analytic aspect of the Strominger-Yau-Zaslow conjecture. In the study of these problems, the focus is on the regularity of solutions of the relevant degenerate and singular equations, especially asymptotic behaviors of solutions near the set of degeneracy. These studies are advanced, challenging and original.

上世纪70年代以来，Yamabe问题、正质量猜想和Calabi猜想等一系列重要的几何问题通过研究相应的非线性椭圆型方程得到了解决。随着（严格）椭圆型方程理论的完善，与几何和物理有关的退化与奇异椭圆型方程引起了广泛的关注，吸引了一批优秀数学家的探索和钻研。本项目合作研究Monge-Ampère方程、极小曲面方程边界爆破问题和边界退化问题, 以及Schwarzschild 空间中波方程解的渐近行为；环簇和齐次toric丛上极致Kähler度量的存在性，多面体上四阶Abreu方程的可解性；退化Monge-Ampère方程和special Lagrangian方程在半空间中解的分类；与Strominger-Yau-Zaslow猜想有关的分析问题，建立与之相关的理论。这些研究主要解决相关退化与奇异方程解的正则性，特别是解在退化集合附近的渐近行为，在退化偏微分方程领域中极富前沿性、挑战性和创新性。

几何分析在几何学的研究中起了重要的作用，它的兴起是在上世纪七十年代中后期和八十年代初期。在此期间，一系列与非线性椭圆型方程相关的重要几何问题得到了解决。随着（严格）椭圆型方程理论的完善，退化椭圆型方程已成为偏微分方程中重要的和活跃的课题之一，特别是与几何和物理有关的退化椭圆型方程更是引起了广泛的关注。由于退化性的多样化，到目前为止退化椭圆型方程并没有完整的理论，与之相关的一些重要问题也没有得到完全解决。本项目着重研究与退化椭圆型方程相关的几类重要几何问题。这些问题的共同特点如下：方程本身定义在紧带边流形上，方程在内部为严格椭圆，退化只发生在边界。本项目主要研究这些问题的解在边界附近行为。这其中的一个重要课题是环簇和齐次toric丛上极值Kahler度量的存在性。项目组在二维情形完整地解决了这一问题。