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的几何 黎曼流形和CR流形 具体来说，他会找到爱因斯坦 具有规定保形结构的球的度量. 无穷大的领域，并解决CR Yamabe问题， 通过发展“CR极小曲面”理论，对三维曲面进行了研究。" 在两个相关的项目中，他将证明球形的存在。 CR结构，并确定是否 带边界黎曼流形的Dirichlet-Neumann映射 确定直到等距的度量。 几何学家认为球体是正弯曲的。 的 在一些实施例中，内管的环面或皮肤是正弯曲的。 部分，但在其他负弯曲。 通过平滑地变形 根据距离的概念，环面可以是平坦的，也可以是零 曲率 两孔圆环可以变形，直到它们 常负曲率 主要研究者将 解决三维Yamabe问题。 因此他 将使具有负欧拉特征的三维流形变形，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.