Mathematical Sciences: Evolution Equations in Geometry

理查德·汉密尔顿将继续他在微分几何中关于热方程的激动人心的工作。在最近的工作中，他证明了在具有正曲率算子的紧致四个流形上，度量演化为常正截曲率的度量。在与Michael Gage的合作中，他证明了平面上的凸曲线在曲线缩短到一点的情况下收缩，一直保持凸性，并在接近末端时变得渐近圆形。在继续这项工作时，他将考虑高维曲率算子的收敛。类似地，他将在初始度量具有负曲率的情况下研究两个球面上的度量的收敛。显然有许多途径需要探索，取得重大成就的可能性似乎很高。

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.