Parabolic problems in conformal geometry

主要研究微分几何中出现的椭圆型和抛物型问题。一个例子是黎曼度规的共形变形（即所谓的Yamabe流）。一个有趣的问题是，解是否接近常数标量曲率的度规，或者流是否会形成奇点。（答案在3 - 5维度是已知的，但一般情况还没有完全理解。）这个问题的解决方案可能需要很好地理解爆炸点附近解的局部性质。另一个可能相关的工具是广义相对论中的正质量定理。同样有趣的是，看看类似的想法是否能在更高维度上阐明规定的标量曲率问题。该计划涉及三个数学领域：微分几何、非线性偏微分方程和变分学。该项目的目标之一是通过曲率研究流形的变形，并更好地理解解的长期行为。其中一些几何演化方程与应用数学中所研究的方程密切相关，如多孔介质方程。

The principal investigator proposes to study certain elliptic and parabolic problems that arise in differential geometry. One example is the conformal deformation of Riemannian metrics by their scalar curvature (the so-called Yamabe flow). An interesting question is whether the solution approaches a metric of constant scalar curvature or whether the flow may develop a singularity. (The answer is known in dimensions 3 -- 5, but the general case is not yet fully understood.) A solution to this problem will likely require a good understanding of the local properties of the solution in a neighborhood of a blow-up point. Another tool that is likely to be relevant is the positive mass theorem in general relativity. It would also be interesting to see whether similar ideas can shed light on the prescribed scalar curvature problem in higher dimensions. The proposed project lies at the intersection of three areas of mathematics: differential geometry, nonlinear partial differential equations, and the calculus of variations. One of the goals of this project is to study the deformation of manifolds by their curvature, and to gain a better understanding of the longtime behavior of the solutions. Some of these geometric evolution equation are closely related to equations studied in applied mathematics, such as the porous medium equation.