Nonlinear Problems in Geometry

主要研究人员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的椭圆型方程，如Monge-Ampere方程或平均曲率方程有很强的联系。特别是，PI将尝试将他对双曲空间中完备的常曲率超曲面的研究，特别是在凸的情况下，推广到常曲率的单连通Lorentz空间形式，即Minkowski空间，de Sitter空间和anti-De Sitter空间。PI还将研究半线性椭圆型偏微分方程组正解的有界性和单调性，特别是在所谓的超临界情况下。全局有界性一般是不成立的，但PI将尝试在边界的一致邻域内显示解的单调性。他还将探索与之密切相关的半稳定解的一致有界性问题。这一问题与稳定极小超曲面的公开Bernstein问题有着密切的联系。最后，PI对Mabuchi，Semmes和Donaldson所描述的完全简并的复Monge-Ampere方程感兴趣，该方程是在固定Kahler类的Kahler势空间中的测地线研究中产生的。该项目旨在开发新的分析技术来解决几何、物理、生物和天文学中的问题，其中基本的物理和几何方程可以用椭圆型偏微分方程来描述。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上取得了极大的成功。这些方法在纯数学和应用数学中有着广泛的应用，特别是量子物理和宇宙学、图像处理、最优设计和计算生物学。

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.