Fully Nonlinear and Higher Order Equations in Geometry

主要研究者：Matthew Gursky，Notre Dame UniversityDMS-0200646完全非线性和高阶几何方程摘要。 在这个建议中所描述的工作在于三个领域的交叉点：高阶椭圆偏微分方程，完全非线性方程，微分几何。 我们研究的方程是几何原点的，由Weyl-Schouten张量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。 在这个问题和在三维空间中规定曲面曲率的更经典的问题之间有很强的结构相似性。 为了分析我们的方程，我们使用了完全非线性和高阶椭圆方程领域的技术。 几何结果在低维中最有趣：例如，我们已经发展了一种技术，用于构造大的共形流形族，这些共形流形允许具有正Ricci曲率的度量。几何学和分析学的相互作用至少可以追溯到十八世纪，几何学是数学研究的一个重要和高度活跃的领域。经典的几何学是从我们希望了解几何学的某些性质发展而来的。微分几何是为了理解弯曲空间的几何而发展起来的，例如，地球表面的曲率，或者广义相对论预测的物质空间的曲率。正如笛卡尔认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技巧来研究，尤其是微分方程。 本研究涉及黎曼几何和共形几何中的几个问题，这些问题的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.