Mathematical Sciences: Evolution Equations in Geometry

数学科学：几何演化方程

• 批准号: 8701613
• 负责人: Burton Rodin
• 金额: $ 17.01万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701613&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Evolution; Equations; Geometry

Richard Hamilton will continue his exciting work on the heat equation in differential geometry. In recent work he has shown that on a compact four manifold with positive curvature operator the metric evolves to one of constant positive sectional curvature. In collaboration with Michael Gage, he showed that a convex curve in the plane shrinks under curve shortening to a point, remaining convex all the time and becoming asymptotically circular near the end. In continuing this work he will consider convergence of curvature operators in higher dimensions. Similarly he will investigate convergence of metrics on the two sphere in the case where the initial metric has some negative curvature. There are clearly many avenues to be explored and the probability of significant achievements seems very high.

理查德·汉密尔顿将继续他在微分几何中的热方程方面令人兴奋的工作。在最近的工作中，他证明了在具有正曲率算子的紧化四流形上度规演化为具有常正截面曲率的度规。他与迈克尔·盖奇（Michael Gage）合作，证明了平面上的凸曲线在曲线缩短到一个点时收缩，一直保持凸，并在接近终点时变为渐近圆形。在继续这项工作中，他将考虑高维曲率算子的收敛性。同样地，他将研究在初始度规有负曲率的情况下，两个球上度规的收敛性。显然有许多途径有待探索，取得重大成就的可能性似乎很高。