Nonlinear Problems in Geometry

主要研究者建议研究洛伦兹和黎曼几何中的一些当前问题，这些问题与完全非线性椭圆方程（如Monge-Ampere方程或平均曲率方程）有很强的联系。特别是PI将试图扩大他的研究完成超曲面的常曲率双曲空间，特别是在凸的情况下，常曲率简单连接洛伦兹空间形式，即闵可夫斯基空间，德西特空间和反德西特空间。PI还将研究半线性椭圆型方程正解的有界性和单调性，特别是在所谓的超临界情况下。整体有界性一般失败，但PI将试图在边界的均匀邻域中显示解的单调性。他还将探讨半稳定解的一致有界性的密切相关的问题。这个问题与稳定极小超曲面的开伯恩斯坦问题有密切的联系。最后，PI对Mabuchi，Semmes和唐纳森描述的完全退化的复Monge-Ampere方程感兴趣，该方程是在固定Kahler类中Kahler势空间的测地线研究中出现的。该项目旨在开发新的分析技术来解决几何，物理，生物学和天文学，其中基本的物理和几何方程可以由椭圆偏微分方程描述。这种技术在模拟曲率现象或使用曲率流作为分析工具的问题中非常成功。这些方法在理论数学和应用数学，特别是量子物理和宇宙学、图像处理、优化设计和计算生物学中有着广泛的应用。

The principal investigator proposes to study a number of current problems in Lorentzian and Riemannian geometry that have a strong connection with fully nonlinear elliptic equations such as Monge-Ampere equations or mean curvature equations. In particular the PI will attempt to extend his study of complete hypersurfaces of constant curvature in hyperbolic space, especially in the convex case, to the constant curvature simply connected Lorentz space forms, namely Minkowski space, de Sitter space and anti-de Sitter space. The PI will also study boundedness and monotonicity properties of positive solutions of semilinear elliptic pde's, especially in the so-called supercritical case. Global boundedness fails in general but the PI will attempt to show monotonicity of solutions in a uniform neighborhood of the boundary. He will also explore the closely related problem of uniform boundedness of semi-stable solutions. There is a close connection of this problem with the open Bernstein problem for stable minimal hypersurfaces. Finally the PI is interested in the totally degenerate complex Monge-Ampere equation described by Mabuchi, Semmes and Donaldson, which arises in the study of geodesics in the space of Kahler potentials in a fixed Kahler class.This project is aimed at developing new analytic techniques to solve problems in geometry, physics, biology and astronomy where the underlying physical and geometrical equations can be described by elliptic partial differential equations. Such techniques have been extremely successful in problems which model curvature phenomena or use curvature flows as an analytic tool. These methods have broad applications in pure and applied mathematics, especially quantum physics and cosmology, image processing, optimal design and computational biology.