Variational Problems and Nonlinear Equations in Geometry

该奖项支持的研究活动是在几何分析领域，所提出的具体问题是在三个领域的交叉点：偏微分方程，微分几何和数学物理。 例如，椭圆算子的泛函行列式是一个起源于谱理论和数学物理的问题，而对所考虑的特定问题的分析是一个导致四阶椭圆方程的变分问题。 相应的拉格朗日量是无界的，解的存在性及其定性性质是高度非平凡的。 类似的方程被用来模拟薄膜的性质。 提出的另一个问题是利用手术方法构造四维流形上的Bach平坦度量。 巴赫张量出现在广义相对论和共形几何中，巴赫平坦条件是黎曼度量空间上曲率为二次的泛函的欧拉-拉格朗日方程，可以看作是具有一定不变性的非线性偏微分方程组。 我们提出的最后一组问题涉及非线性方程的共形几何和理论的最佳运输。 我们特别感兴趣的是这些方程的奇异解的构造，以此来研究具有规定曲率条件的完备黎曼度量的存在性。几何与分析的相互作用至少可以追溯到世纪，至今仍是数学研究中一个重要而活跃的领域。经典的几何学是从我们想要了解物理世界的某些性质的愿望中发展出来的，微分几何是为了理解弯曲空间的几何-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。在同样的方式，笛卡尔意识到，平面几何可以研究使用代数，所以微分几何可以研究使用技术分析，特别是微分方程。在这个建议中的研究涉及几何和数学物理的不同问题，但通过数学分析在其研究中所发挥的作用而团结起来。

The research activities supported by this award are in the field of geometric analysis, and the specific problems posed are at the intersection of three fields: partial differential equations, differential geometry, and mathematical physics. For example, the functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem considered is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another problem proposed is the construction of Bach-flat metrics on four-manifolds using surgery methods. The Bach tensor arises in general relativity and conformal geometry, and the Bach-flat condition is the Euler-Lagrange equation for a functional on the space of Riemannian metrics which is quadratic in the curvature, and can be viewed as a nonlinear system of partial differential equations with certain invariance properties. The final set of problems we propose involves nonlinear equations arising in conformal geometry and the theory of optimal transportation. We are specifically interested in the construction of singular solutions of these equations as a way to study the existence of complete Riemannian metrics with prescribed curvature conditions. 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 as 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 disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study.