Mathematical Sciences: Partial Differential Equations in Riemannian and CR Geometry

首席研究员将研究线性和非线性偏微分方程在黎曼流形和CR流形几何中的应用。具体来说，他将在球面无穷远处找到具有规定保形结构的球的爱因斯坦度量，并通过发展“CR最小曲面”理论解决三维空间的CR Yamabe问题。在两个相关的项目中，他将证明某些3-流形上球形CR结构的存在性，并确定具有边界的黎曼流形的Dirichlet-to-Neumann映射是否决定了等距度规。几何学家认为球体是正弯曲的。环面，或内管的皮肤，在某些部分是正弯曲的，但在其他部分是负弯曲的。通过平滑地变形距离的概念，环面可以变得平坦或具有零曲率。双孔环面可以变形，直到它们具有恒定的负曲率。首席研究员将找到三维山边问题的解决方案。因此，他将具有负欧拉特性的3流形变形，直到它们具有恒定的负曲率。

The principal investigator will study applications of linear and nonlinear partial differential equations to the geometry of Riemannian and CR manifolds. Specifically, he will find Einstein metrics of the ball with prescribed conformal structure at infinity on the sphere, and solve the CR Yamabe problem in dimension three by developing a theory of "CR minimal surfaces." In two related projects he will prove the existence of spherical CR structures on certain 3-manifolds, and determine whether the Dirichlet-to-Neumann map of a Riemannian manifold with boundary determines the metric up to isometry. Geometers consider spheres to be positively curved. The torus, or skin of an inner tube, is positively curved in some parts but negatively curved in others. By smoothly deforming the concept of distance, the torus can be made flat or have zero curvature. Two-holed tori can be deformed until they have constant negative curvature. The principal investigator will find a solution to the three-dimensional Yamabe problem. Thus he will deform 3-manifolds with negative Euler characteristic until they have constant negative curvature.