Parabolic problems in conformal geometry

主要研究者建议研究微分几何中出现的某些椭圆和抛物问题。一个例子是黎曼度量的标量曲率的共形变形（所谓的Yamabe流）。一个有趣的问题是，该解是否接近常数标量曲率的度量，或者流动是否可能发展为奇点。(The答案在3 - 5维中是已知的，但一般情况尚未完全理解。这个问题的解决方案可能需要很好地理解解在爆破点附近的局部性质。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阐明高维中的规定标量曲率问题。该项目的建议在于三个数学领域的交集：微分几何，非线性偏微分方程，变分法。这个项目的目标之一是研究流形的曲率变形，并更好地理解解的长期行为。这些几何演化方程中的一些与应用数学中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.