Nonlinear Problems in Geometry

首席研究员将以一种新颖的方式研究微分几何和黎曼几何中与平均曲率方程相关或与完全非线性椭圆方程（如蒙日-安培方程）有密切联系的一些经典问题。这些包括规定的平均曲率方程和更一般的黎曼流形图的曲率函数，以及在无限远处具有规定边界的双曲空间中具有恒定平均曲率的超曲面。我们还应用完全非线性椭圆偏微分方程的方法，研究了从下估计Cartan-Hadamard流形中凸超曲面的绝对高斯曲率的几何问题。我们继续研究在涉及曲率的问题中自然出现的完全非线性椭圆方程理论的几何方面。这些方程是最有用的，也是最难研究的，在纯数学和应用数学中有着广泛的应用，特别是在图像处理、优化设计、计算生物学和数学物理中。例如，在图像处理和脑配准问题中使用平均曲率流和其他曲率流现在是相当标准的。同样，我们期望我们目前的研究在未来成为一种标准工具。

The principal investigator will study a number of classical problems in Differential and Riemannian geometry that are related to mean curvature equations or have a strong connection with fully nonlinear elliptic equations such as Monge-Ampere equations in some novel way. These include the equations of prescribed mean curvature and more general curvature functions of graphs in Riemannian manifolds MxR and hypersurfaces of constant mean curvature in hyperbolic space with prescribed boundary at infinity. We also apply the methods of fully nonlinear elliptic pde to study the geometric problem of estimating from below, the total absolute Gauss curvature for convex hypersurfaces in a Cartan-Hadamard manifold.We continue to study the geometric aspects of the theory of fully nonlinear elliptic equations that arise naturally in problems involving curvature. These equations are the most useful and the most difficult to study and are broadly applicable in pure and applied mathematics, especially in image processing, optimal design, computational biology and mathematical physics. For example, the use of mean curvature flow and other curvature flows in image processing and problems of brain registration is now quite standard. We similarly expect that our current research will become a standard tool in the future.