Nonlinear Problems in Geometry

本文研究了双曲空间（具有给定的有限边界或渐近边界）中常曲率和规定曲率超曲面的存在性问题，以及Minkowski空间中类空间超曲面的相应问题。在涉及的许多技术挑战中，需要一个最大主曲率的一般最大原理。我们还打算研究黑洞和宇宙尘埃的模型，其中选择合适的度量将爱因斯坦-麦克斯韦方程简化为单个半线性椭圆方程。除了许多微妙的存在问题外，对由此产生的时空的几何性质的分析还导致了许多有趣的问题。许多不同的兴趣领域，如图像处理和医学成像，优化设计，计算生物学和宇宙学利用模型，或显式或隐式地涉及非线性椭圆方程，描述曲率或曲率流（甚至更复杂的过程）。例如，25年前，水平集平均曲率流作为图像处理的计算工具的新颖使用需要更好的理论理解，这导致了新的数学突破。今天，潜在的模型和问题要复杂得多，并强调需要对所涉及的数学有更好的理论理解。

We propose to study questions of existence of hypersurfaces of constant and prescribed curvature in hyperbolic space (with a given finite or asymptotic boundary) and the corresponding problems for spacelike hypersurfaces in Minkowski space. Among the many technical challenges involved is the need for a general maximum principle for the maximum principle curvature. We also intend to study models of black holes and cosmological dust where the choice of a suitable metric reduces the Einstein-Maxwell equations to a single semilinear elliptic equation. In addition to many subtle existence problems, the analysis of the geometric properties of the resulting spacetime leads to many interesting questions.Many diverse areas of interest, such as image processing and medical imaging , optimal design, computational biology and cosmology utilize models that either explicitly or implicitly involve the nonlinear elliptic equations that describe curvature or curvature flows (and even more complicated processes). For example, twenty five years ago the novel use of the level set mean curvature flow as a computational tool in image processing required a better theoretical understanding which led to new mathematical breakthroughs. Today, the underlying models and problems are much more sophisticated and underscore the need for a better theoretical understanding of the mathematics involved.