Mathematical Sciences: Evolution Equations in Geometry

主要研究者将研究一种方法来证明所有具有正标量曲率的闭3流形都是通过执行连通和从球面空间形式得到的。本文将分析具有正各向同性曲率的4-流形的一个类似问题。该项目的其他部分包括对四维schoen苍蝇猜想的研究，以及对Kaehler和辛结构的研究。球面上没有任何中间交点的闭合曲线将平面分成两个圆盘。类似地，在普通的三维空间中，一个适当光滑的球体的边界是一个球，如果无限，它的边界是第二个球，它坍缩成一个点。主要研究者将研究一种利用热流来解决四维空间中三维光滑球体的类似问题的方法。

The principal investigator will study a method for showing that all closed 3-manifolds with positive scalar curvature are obtained from spherical space forms by performing connected sums. An analogous problem for 4-manifolds with positive isotropic curvature will be analyzed. Other parts of the project include investigations into the four-dimensional Schoenflies conjecture, as well as studies of Kaehler and symplectic structures. A closed curve on a sphere, which does not have any intermediate intersections, separates the plane into two disks. Analogously, a properly smooth sphere lying in ordinary 3-space bounds one ball, and bounds a second ball if infinity is collapsed to a point. The principal investigator will study a method using heat flow to solve the analogous problem for three- dimensional smooth spheres lying in four-dimensional space.