Partial Differential Equations on Complex and Symplectic Manifolds

PI提出的研究集中在与复流形和辛流形的几何有关的几个基本问题上，这些问题可以用非线性偏微分方程组来研究。在第一个项目中，PI将研究Donaldson最近的一个猜想，该猜想旨在将Kahler几何中的Yau定理推广到辛四流形，建立在他与Weinkove和Yau的工作基础上。如果这一猜想成立，将为构造紧辛四维流形上的辛型提供一个强有力的新工具，并将在辛拓扑上有显著的应用。第二个项目涉及紧致Calabi-Yau流形的几何，特别是当Calabi-Yau流形上的Ricci-Flat Kahler度量的上同调类接近Kahler锥的边界时，它们的退化方式。弦理论家也研究了这些简并与镜像对称性有关的问题。PI建议继续他对这些退化的研究，以及在接近一个大型复杂结构极限的五次三重数族上调查Ricci平坦度量。第三个项目属于紧致Kahler流形上的正则度量领域，例如Kahler-Einstein或常标量曲率Kahler度量。人们认为这种正则度量的存在应该等价于流形的代数稳定性。PI将使用与这些问题相关的两个自然演化方程Kahler-Ricci流和Calabi流来研究这一问题，目的是通过使用自然能量泛函将流的极限行为与代数稳定性联系起来。最后一个项目还涉及到正则Kahler度量，更具体地说，是关于上同调类中具有充足正则丛的复杂曲面上常标量曲率Kahler度量的存在性问题。我们将考虑的大多数问题，例如爱因斯坦方程，最初是由物理学家在寻找自然基本定律的模型时发现的。最近，与拟议的研究密切相关的几何方面在高能物理中得到了应用，并被用来加深我们对宇宙和基本粒子的理解。PI的研究的几何思想围绕着寻找几何空间的最佳形状的问题，即具有最大可能对称性的几何空间，并理解在不存在这种最佳形状的空间中可能形成的奇点。在这些问题上的任何进展不仅将为数学中的一些基本问题提供一些线索，而且还将在物理学和其他科学中得到应用。

The PI proposed research focuses on several basic problems related to the geometry of complex and symplectic manifolds, which can be studied using nonlinear PDEs. In the first project the PI will study a recent conjecture of Donaldson that aims at extending Yau's theorem in Kahler geometry to symplectic four-manifolds, building on his work with Weinkove and Yau. If proved, this conjecture would provide a powerful new tool to construct symplectic forms on compact symplectic four-manifolds, and would have striking applications to symplectic topology. The second project regards the geometry of compact Calabi-Yau manifolds, and specifically the way in which Ricci-flat Kahler metrics on a Calabi-Yau manifold can degenerate when their cohomology class approaches the boundary of the Kahler cone. These degenerations have also been studied by string theorists in connection with mirror symmetry. The PI proposes to continue his study of these degenerations, as well as investigating Ricci-flat metrics on a family of quintic threefolds near a large complex structure limit. The third project falls in the area of canonical metrics on compact Kahler manifolds, such as Kahler-Einstein or constant scalar curvature Kahler metrics. It is believed that the existence of such canonical metrics should be equivalent to the algebraic stability of the manifold. The PI will study this using two natural evolution equations associated to these problems, the Kahler-Ricci flow and the Calabi flow, with the aim of connecting the limiting behaviour of the flows to algebraic stability through the use of natural energy functionals. The final project also involves canonical Kahler metrics, and more specifically the problem of existence of constant scalar curvature Kahler metrics on complex surfaces with ample canonical bundle in cohomology classes that are known to be stable.Most of the problems that we will consider, for example the Einstein equations, were originally discovered by physicists who were searching for models of the fundamental laws of nature. More recently, geometric aspects closely related to the proposed research have found applications in high energy physics, and are being used to deepen our understanding of the Universe and of elementary particles. The geometric ideas of the PI's research revolve around the problem of finding the optimal shape of a geometric space, the one with the largest possible symmetry, and understanding the possible singularities that form in spaces where such an optimal shape does not exist. Any progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.