Fully Nonlinear Partial Differential Equations in Geometry

AbstractAward：DMS-0505632项目负责人：关博本项目的主要目的是了解微分几何中出现的一些非线性椭圆和抛物方程。 研究将集中在三种类型的几何问题：高原型问题的超曲面在欧几里德空间和更一般的流形;闵可夫斯基型问题的发现封闭凸超曲面的规定温加滕曲率，和问题的等距嵌入。 这些问题中的每一个都涉及到求解一些完全非线性偏微分方程。由于它们起源于几何学，这些方程密切相关，因此有许多共同的技术问题。 因此，其中一个问题的进展很可能会导致其他问题的进展，更一般地说，在非线性偏微分方程理论（流形上）和几何分析中，完全非线性方程也出现在其他问题中，如几何学和物理学以及科学和工程学中。我们的研究结果可能会对物理学中的弦理论和最优控制理论等领域产生影响和应用。

AbstractAward: DMS-0505632Principal Investigator: Bo GuanThe primary goal of this project is to understand some nonlinearelliptic and parabolic equations which arise from problems indifferential geometry. The research will be focused on threetypes of geometric problems: Plateau-type problems forhypersurfaces in Euclidean space and in more general manifolds;the Minkowski-type problems of finding closed convexhypersurfaces of prescribed Weingarten curvature, and problems inisometric embedding. Each of these problems involves solvingsome fully nonlinear partial differential equations. Because oftheir origins in geometry, these equations are closely relatedand therefore share many common technical issues. Thus, progressin one of the problems will very likely lead to progress in theothers, and more generally in the theory of nonlinear partialdifferential equations (on manifolds) and geometric analysis.Fully nonlinear equations also arise in other problems ingeometry and physics as well as in science andengineering. Results from our research may have impact andapplications in areas such as string theory in physics, andoptimal control theory.