Differential Equations and the Geometry of Manifolds

本课题研究的一个重要动机是了解空间的几何和拓扑之间的关系。后者，拓扑学，是对空间在连续拉伸或弯曲下不变的性质的研究，而前者，几何学，涉及到对距离的理解，并且更加刚性。例如，我们的地球表面是一个球体，人们通过计算沿大圆路径的长度来测量球面上两点之间的距离。（地球实际上是一个扁球体，但它非常接近完美的球体。）你可以想象通过推或拉小或大的区域来扭曲地球的几何形状。这种变形的形状在某种程度上不如我们熟悉的圆形地球那么吸引人，而且有很多方法可以使这种“变形”的概念非常精确，因为可以最小化某种总能量测量。这与物理原理直接相关，即物理系统的状态将趋向于使总能量最小的最终构型。这个想法也可以推广到被称为流形的高维物体上，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，我们就在一个四维的宇宙中。）为了理解这些高维物体，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。该方案中的项目是在这样的空间上定义适当的能量，并寻找重要的最佳几何形状，使总能量最小化。用更专业的术语来说，PI的研究，从广义上讲，是使用几何起源的偏微分方程的解来研究可微流形的性质。PI的主要研究领域是爱因斯坦轨道的非广域化，K3曲面上塌缩ricci平坦度量序列的构造，标量平坦Kahler ALE度量的全局模空间的构造，以及轨道Yamabe问题的研究。在与Morteza的共同工作中，在渐近双曲Einstein环境下证明了爱因斯坦度量的存在性定理，推广了四维Biquard的结果，并对该工作进行了扩展和推广。在与Hein， Sun和Zhang正在进行的工作中，PI已经构建了K3表面上的里奇平面度量的新例子，这些曲面坍缩到一个区间，海森堡零流形作为纤维出现在规则的坍缩区域。这项工作产生了许多有趣的问题，特别是将这些退化与K3表面的极化退化联系起来。在与Han的合作中，PI提出了一个在无穷远处为某些群构造标量平坦Kahler ALE度量的全局模空间的计划。此外，在与Tao Ju的合作中，PI已经证明了轨道Yamabe问题的一些不存在性结果，并计划进一步推广这一结果。最后，PI致力于整合研究和教育，培养多层次的智力发展。PI一直积极地在数学界外展和组织会议。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

An important motivation for the research in this project 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 between points on it by computing lengths along great circle paths. (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 deformed shape is somehow less appealing than the familiar round Earth, and there are many ways to make this notion of being "bent out of shape" 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 also be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of 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 in this proposal are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy.In more technical terms, the research of the PI is, broadly speaking, to use solutions of partial differential equations which are geometric in origin to study properties of differentiable manifolds. The main areas of concentration of the PI's research are the desingularization of Einstein orbifolds, the construction of sequences of collapsing Ricci-flat metrics on K3 surfaces, the construction of a global moduli space of scalar-flat Kahler ALE metrics, and the study of the orbifold Yamabe problem. In joint work with Morteza, an existence theorem for Einstein metrics was proved in the asymptotically hyperbolic Einstein setting, which generalized a result of Biquard in dimension four, and the PI proposes several extensions and generalizations of this work. In ongoing work with Hein, Sun, and Zhang, the PI has constructed new examples of Ricci-flat metrics on K3 surfaces which collapse to an interval, with Heisenberg nilmanifolds occurring as fibers in the regular collapsing regions. There are many interesting questions resulting from this work, especially to relate these degenerations to polarized degenerations of K3 surfaces. In joint work with Han, the PI proposes a plan towards constructing a global moduli space of scalar-flat Kahler ALE metrics for certain groups at infinity. Also, in joint work with Tao Ju, the PI has proved some nonexistence results for the orbifold Yamabe problem, and plans to generalize this further. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels. The PI has been active in outreach and organization of conferences in the mathematics community.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.