Fully Nonlinear and Higher Order Equations in Geometry

PI:马修•古尔斯基圣母universitydms - 0200646 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 在geometryAbstract完全非线性和高阶方程。本提案中所描述的工作涉及三个领域的交叉：高阶椭圆偏微分方程、全非线性方程和微分几何。我们研究的方程是几何起源的，由Weyl-Schouten张量的特征值的初等对称多项式给出，特别是在度规的共形变形下。这个问题与更经典的在三维空间中规定曲面曲率的问题在结构上有很强的相似性。为了分析我们的方程，我们使用了全非线性和高阶椭圆方程领域的技术。几何结果在低维中是最有趣的：例如，我们已经开发了一种构造大族共形流形的技术，这些流形允许具有正维曲率的度量。几何和分析的相互作用至少可以追溯到18世纪，并且仍然是数学研究的重要和高度活跃的领域。经典的几何学科源于我们想要了解物理世界的某些特性的愿望，而微分几何的发展是为了理解弯曲空间的几何——例如，地球表面的曲率，或者广义相对论所预测的物质对空间的曲率。就像笛卡尔意识到平面几何可以用代数来研究一样，微分几何也可以用分析的方法来研究，尤其是微分方程。本提案中的研究涉及黎曼几何和共形几何中的几个此类问题，其分析需要数学分析中各个领域的技术。

PI: Matthew Gursky, Notre Dame UniversityDMS-0200646%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Fully nonlinear and higher order equations in geometryAbstract. The work described in this proposal lies at the intersection of three fields: higher order elliptic partialdifferential equations, fully nonlinear equations, and differential geometry. The equations we study are geometricin origin, and given by elementary symmetric polynomials of the eigenvalues of the Weyl-Schouten tensor, specificallyunder conformal deformations of the metric. There is astrong structural analogy between this problem and the moreclassical problem of prescribing the curvature(s) of a surface in three-dimensional space. To analyze our equationswe use techniques from the field of fully nonlinear and higher order elliptic equations. The geometric consequencesare most interesting in low dimensions: for example, wehave developed a technique for constructing large families of conformal manifolds which admit metrics with positiveRicci curvature,The interaction of geometry and analysis dates backto at least the eighteenth century, and yet continues to bean important and highly active field of mathematical research.The classical subject of geometry grew out of our desire tounderstand certain properties of the physical world, anddifferential geometry was developed to understand the geometry of curved spaces--for example, the curvatureof the surface of the earth, or the curvature of space by matter predicted by general relativity. In the same way that Descartes realized that planegeometry can be studied using algebra, so differentialgeometry can be studied using techniques from analysis,especially differential equations. This research in thisproposal involves several such problems from Riemmanianand conformal geometry, whose analysis requires techniquesfrom various fields within mathematical analysis.