Differential Equations and the Geometry of Manifolds

微分方程和流形几何

• 批准号: 1405725
• 负责人: Jeff Viaclovsky
• 金额: $ 35.13万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405725&HistoricalAwards=false
• 关键词: Differential; Equations; Geometry; Manifolds

An important motivation for the research of the PI 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 than 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 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 existence of critical metrics generalizing the Einstein condition, the study of quadratic curvature functionals on Riemannian manifolds, the study of critical ALE metrics and orbifolds, and properties of moduli spaces of critical metrics. In joint work with Matt Gursky, the PI has proved existence of critical metrics on various four-manifolds. The proposed research is to further study the properties of the moduli space of such solutions. This is related to orbifold compactness theorems previously studied by Tian-Viaclovsky. The PI has previously demonstrated non-solvability of the Yamabe problem on certain compact orbifolds, which showed that the orbifold Yamabe problem is more subtle than in the case of smooth manifolds. The PI proposes further exploration of this phenomenon, and of connections with the notion of mass of ALE spaces. There has been a considerable amount of research on the existence of anti-self-dual metrics on compact manifolds; they have been shown to exist in abundance. Another goal of the proposal is therefore to understand global properties of the moduli space in certain cases; especially for orbifold-cone anti-self-dual metrics. 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.

PI研究的一个重要动机是理解空间的几何和拓扑之间的关系。后者，拓扑学，是研究空间的性质，这些性质在空间的连续拉伸或弯曲下是不变的，而前者，几何学，涉及理解距离，并且更加严格。例如，地球的表面是一个球体，人们通过计算大圆的弧长来测量它的距离（地球实际上是一个扁球体，但它非常接近完美的球形）。你可以想象通过向内推或拉小的或大的区域来扭曲几何形状，从而使地球变形。这样的变形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个概念非常精确，以最小化某种总能量测量。这与物理原理直接相关，物理原理说，物理系统的状态将趋向于使总能量最小化的最终配置。这个想法可以推广到更高维的物体，称为流形，这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。为了理解这些类型的高维物体，人们试图找到最好的方法来测量它们的距离，使用最少的能量，并最大限度地提高空间的对称性。该计划的目标是在这样的空间上定义适当的能量，并寻找使总能量最小的重要的最优几何。用更专业的术语来说，PI的研究是，广义地说，使用偏微分方程的解，这是几何起源，以研究可微流形的性质。主要领域的浓度PI的研究是存在的临界度量推广爱因斯坦条件，研究二次曲率泛函的黎曼流形，研究临界ALE度量和orbifolds，和性质的模空间的临界度量。在与Matt Gursky的合作中，PI证明了各种四维流形上临界度量的存在性。本文的主要目的是进一步研究这类解的模空间性质。这与田-维亚克洛夫斯基以前研究的轨道紧性定理有关。PI之前已经证明了Yamabe问题在某些紧致轨道上的不可解性，这表明轨道Yamabe问题比光滑流形的情况更加微妙。PI建议进一步探索这一现象，以及与ALE空间质量概念的联系。关于紧致流形上的反自对偶度量的存在性已经有了相当多的研究;它们已经被证明是大量存在的。因此，该提案的另一个目标是理解某些情况下模空间的全局性质，特别是对于轨道锥反自对偶度量。最后，PI致力于整合研究和教育，并在多个层面上培养智力发展。PI一直积极参与数学界的外联和会议组织。