The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world, and differential geometry was developed to understand the geometry of curved spaces--for example, the curvature of the surface of the earth, or the curvature of space by matter predicted by general relativity. In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this proposal involves several such problems from Riemmanian and conformal geometry, whose analysis requires techniques from various fields within mathematical analysis.
The specific problems we pose are at the intersection of three fields: higher order elliptic partial differential equations, fully nonlinear equations, and differential geometry. The equations we study are geometric in origin, and given by symmetric functions of the eigenvalues of the Ricci tensor, specifically under conformal deformations of the metric. There is a strong structural analogy between this problem and the more classical problem of prescribing the curvature(s) of a surface in three-dimensional space. To analyze our equations we use techniques from both the theory of elliptic equations and from comparison geometry. For example, to understand on a microscopic level the blow-up behavior of a sequence of solutions, we are confronted with the problem of understanding the tangent cone at infinity of certain asymptotically flat spaces.
The geometric consequences of these results are most interesting in low dimensions: for example, we have developed a technique for constructing large families of conformal manifolds which admit metrics with positive Ricci curvature.
