Fully Nonlinear and Higher Order Equations in Geometry

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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.