Differential equations and the geometry of manifolds

摘要奖：dms-0804042首席研究员：Jeff A.Viaclovsky该奖项支持的第一个项目涉及四维临界黎曼度量的正则性和体积增长性质，以及在模空间的紧性和临界度量的存在性方面的应用，例如反自对偶和极值Kaehler度量。在一定的几何非折叠假设下，适当的模空间可以通过添加度量或类分叉奇点来紧致。这将爱因斯坦度量的结果推广到没有逐点Ricci曲率界限的度量的情况。一个长期的目标是将紧致性定理扩展到包括折叠的可能性，并找到四流形的微分拓扑的其他应用。第二个项目研究了正交复结构，以及与扭曲空间的子簇之间的关系。相应的方程是共形不变的，因此一个自然的问题是寻找在共形群作用下不变的簇的性质。这对于理解紧致厄米特流形的几何有应用。第三个项目涉及曲率的变形，完全非线性曲率方程解的存在性，以及与三维和四维流形上的黎曼泛函的关系。一个关键的问题是对度量进行共形变形，以规定Ricci张量的本征值的对称函数(推广了Yamabe问题)，并找到自然的共形不变条件，从而使度量可以从较弱的积分Pinch条件变形到较强的逐点Pinch条件。这项研究的一个重要动机是了解空间几何和拓扑之间的关系。后者，拓扑学，是研究空间在连续拉伸或弯曲下不变的性质，而前者，几何学，涉及到距离的理解，更严格。例如，我们星球的表面是一个球体，人们通过计算大圆的弧长来测量其上的距离(地球实际上是一个扁平的球体，但它非常接近完美的球体)。人们可以想象通过推入或拉动小区域或大区域来扭曲几何图形来使地球变形。这样的形变不如人们熟悉的环绕地球的形变那么有吸引力，有很多方法可以让这个概念变得非常精确，从而最小化某种总能量测量。这与物理原理直接相关，物理原理认为，物理系统的状态将趋向于使总能量最小的最终构型。这个想法可以推广到称为流形的更高维的物体上，流形是我们星球表面的广义版本。例如，我们生活的空间是三维的，如果包括时间，我们就处于一个四维的宇宙中。为了理解这些类型的高维物体，人们试图找到最好的方法来测量它们上的距离，使用最少的能量，并最大化空间的对称性。上面描述的项目是在三维和四维空间中定义适当的能量，并寻找使总能量最小的重要的最佳几何形状。

Abstract Award: DMS-0804042 Principal Investigator: Jeff A. ViaclovskyThe first project supported by this award deals with regularity and volume growth properties of critical Riemannian metrics in dimension four, and applications to the compactness of moduli spaces and existence of critical metrics, such as anti-self-dual and extremal Kaehler metrics. With certain geometric noncollapsing assumptions, the appropriate moduli spaces can be compactified by adding metrics orbifold-like singularities. This generalizes results for Einstein metrics to the case of metrics which do not have pointwise Ricci curvature bounds. A long-term goal is to extend the compactness theorem to include the possibility of collapsing, and to find other applications to the differential topology of four-manifolds. The second project deals with orthogonal complex structures, and the relation with subvarieties of twistor spaces. The corresponding equation is conformally invariant, so a natural problem is to find properties of varieties which are invariant under the action of the conformal group. This has applications to understanding the geometry of compact Hermitian manifolds. The third project involves deformation of curvatures, existence of solutions to fully nonlinear curvature equations, and relations with Riemannian functionals on three and four-manifolds. A crucial problem is to conformally deform a metric to prescribe a symmetric function of the eigenvalues of the Ricci tensor (generalizing the Yamabe problem), and to find natural conformally invariant conditions so that a metric can be deformed from a weaker integral pinching condition to a stronger pointwise pinching condition. An important motivation for this research is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing that the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of the our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects described above are to define appropriate energies on such spaces in dimensions three and four, and to seek out the important optimal geometries which minimize the total energy.