Nonlinear Problems in Geometry

几何非线性问题

• 批准号: 0904009
• 负责人: Joel Spruck
• 金额: $ 17.22万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904009&HistoricalAwards=false
• 关键词: Nonlinear; Problems; Geometry

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.

本文研究了双曲空间（具有给定的有限或渐近边界）中常曲率和预定曲率超曲面的存在性问题以及Minkowski空间中类空超曲面的相应问题。在所涉及的许多技术挑战中，需要一个最大主曲率的一般最大值原理。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-麦克斯韦方程组简化为一个 半线性椭圆型方程除了许多微妙的存在性问题外，时空几何性质的分析也引出了许多有趣的问题。许多不同的领域，如图像处理和医学成像、最优设计、计算生物学和宇宙学，都利用了明确或隐含地涉及非线性椭圆方程的模型来描述曲率或曲率流（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中的计算工具的新用途需要更好的理论理解，这导致了新的数学突破。今天，基本的模型和问题更加复杂，并强调需要对所涉及的数学有更好的理论理解。