Nonlinear Problems in Geometry

题目：几何中的非线性问题摘要主要研究者提议研究微分几何和黎曼几何中的一些问题，这些问题可能以某种新颖的方式用完全非线性椭圆方程（如蒙日-安培方程或平均曲率方程）来描述或与它们有很强的联系。这些包括经典的尖锐等周不等式在负曲线曼流形上的扩展，在无限远处具有规定边界的双曲空间中恒定平均曲率的超曲面以及夹紧板基调的最优域。主要研究者的目标是发展基本的几何和解析方法来研究高度非线性问题，这些问题在纯数学和应用数学的几个领域，特别是在微分几何，图像处理，优化设计，磁流体动力学和数学物理中具有重要意义。这些问题是用涉及隐式定义的曲率函数（或动态曲率流，如平均曲率流）的高度非线性偏微分方程来表述的，并且通常是涉及自由边界的变分性质。

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.