Nonlinear Problems in Geometry

主要研究者将研究一些经典的微分和黎曼几何问题，这些问题与平均曲率方程有关，或者与完全非线性椭圆方程（如Monge-Ampere方程）有很强的联系。其中包括黎曼流形MxR中的图的规定平均曲率方程和更一般的曲率函数，以及在无穷远处具有规定边界的双曲空间中的常平均曲率超曲面。我们还应用完全非线性椭圆型偏微分方程的方法来研究Cartan-Hadamard流形中凸超曲面的总绝对Gauss曲率的几何估计问题，我们继续研究自然出现在曲率问题中的完全非线性椭圆型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，在理论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，在图像处理和大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。我们同样希望我们目前的研究在未来成为标准工具。

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.