Variational Problems and Nonlinear Equations in Geometry

该奖项支持的研究活动是几何分析领域，提出的具体问题涉及三个领域：偏微分方程组、微分几何和数学物理。例如，椭圆算子的函数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型方程的变分问题。相关的拉格朗日是无界的，解的存在及其定性性质是非常不平凡的。类似的方程也被用来模拟薄膜的性质。提出的另一个问题是用外科方法构造四维流形上的Bach-Flat度量。Bach张量起源于广义相对论和共形几何，而Bach-Flat条件是曲率二次的黎曼度量空间上泛函的Euler-Lagrange方程，可以看作是一个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保角几何中的非线性方程和最优运输理论。我们特别感兴趣的是这些方程的奇异解的构造，作为研究具有指定曲率条件的完备黎曼度量的存在性的一种方法。几何和分析的相互作用至少可以追溯到18世纪，但仍然是数学研究的一个重要和高度活跃的领域。经典的几何学科源于我们理解物理世界某些属性的愿望，而微分几何的发展是为了理解弯曲空间的几何--例如，地球表面的曲率，或广义相对论预测的由物质引起的空间的曲率。就像笛卡尔意识到平面几何可以用代数来研究一样，所以微分几何也可以用分析中的技巧来研究，特别是微分方程。这项建议中的研究涉及与几何和数学物理截然不同的问题，但由于数学分析在研究中所起的作用而联系在一起。

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.