Fully Nonlinear Partial Differential Equations in Geometry

项目负责人：关博，主要目的是研究微分几何问题中出现的一些非线性椭圆型和抛物型方程。研究将集中在三种类型的几何问题：欧几里得空间和更一般流形中的超曲面的高原型问题；minkowski型求给定Weingarten曲率的闭凸超曲面的问题和等量嵌入问题。这些问题中的每一个都涉及到解决一些完全非线性的偏微分方程。由于它们起源于几何，这些方程是密切相关的，因此有许多共同的技术问题。因此，在一个问题上的进展很可能会导致在其他问题上的进展，更普遍的是在非线性偏微分方程（流形）和几何分析的理论上的进展。完全非线性方程也出现在几何和物理以及科学和工程的其他问题中。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.