Nonlinear Problems in Geometry

提案DMS-0306197 PI：Joel Spruck，约翰霍普金斯大学标题：几何学中的非线性问题 摘要主要研究者提出要研究微分几何和黎曼几何中的一些相关问题，这些问题可以用完全非线性椭圆方程（如Monge-Ampere方程或平均曲率方程）以某种新颖的方式描述，或与之有很强的联系。其中包括将经典的sharp等周不等式推广到负曲黎曼流形，双曲空间中具有预定边界的常平均曲率超曲面以及固支板的基音的最优域.主要研究者的目标是发展基础几何和分析方法来研究在纯数学和应用数学的几个领域中具有重要意义的高度非线性问题，特别是在微分几何，图像处理，优化设计，磁流体力学和数学物理。这些问题都是用高度非线性偏微分方程来表述的，涉及到隐式定义的曲率函数（或动态曲率流，如平均曲率流），而且往往是变分性质的，涉及到自由边界。

PROPOSAL DMS-0306197PI: Joel Spruck, Johns Hopkins UniversityTITLE: NONLINEAR PROBLEMS IN GEOMETRY AbstractThe principal investigator proposes to study a number ofproblems in Differential and Riemannian geometrythat are related in that they may be described by, or have astrong connection with, fully nonlinear elliptic equationssuch as Monge-Ampere equations or mean curvature equations insome novel way. These include extensions of the classicalsharp isoperimetric inequality to negatively curvedRiemannian manifolds, hypersurfaces of constant meancurvature in hyperbolic space with prescribed boundary atinfinity and the optimal domain for the fundamental tone ofa clamped plate.The aim of the Principal investigator is to develop fundamentalgeometric and analytic methods to study highly nonlinear problemsthat are of importance in several fields of pure and applied mathematics,especially in Differential Geometry, image processing, optimal design,magnetohydrodynamics and mathematical physics. These problems areformulated in terms of highly nonlinear PDE's involving implicitly definedfunctions of curvature (or dynamic curvature flows such as mean curvatureflow) and are often variational in nature involving free boundaries.