Mathematical Sciences: Partial Differential Equations in Riemannian and CR Geometry

主要研究人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。具体地说，他将找到球上具有规定的无穷大共形结构的球的爱因斯坦度量，并通过发展“CR极小曲面”理论来解决三维中的CR Yamabe问题。在两个相关的项目中，他将证明某些3-流形上球面CR结构的存在，并确定带边界的黎曼流形的Dirichlet-to-Neumann映射是否决定度量达到等距。几何学家认为球体是正曲线的。圆环，或内管的蒙皮，在某些部分是正弯曲的，在其他部分是负弯曲的。通过平滑地变形距离的概念，可以使圆环变平或具有零曲率。双孔环面可以变形，直到它们具有恒定的负曲率。首席调查员将找到解决三维山部问题的办法。因此，他将变形具有负Euler特征的3-流形，直到它们具有常负曲率。

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.