关于Hermitian Yang-Mills度量的两个问题

Two problems on Hermitian Yang-Mills metrics are concerned. The first one is to understand the limiting behavior of a class of Hermitian Yang-Mills metrics on a stable vector bundle over the complex torus or an elliptic K3 surface. The method is to construct a family of desired Hermitian metrics and then to compare the difference between them and the unknown Hermitian Yang-Mills metrics. In the case of the complex torus, great progress has been made, including the construction of a family of desired Hermitian metrics, the estimate of the lower bound of zero order and the higher order estimates. Now the remaining issue is to estimate the upper bound of zero order. In the case of the elliptic K3 surface, the key issue is to construct a family of Hermitian Yang-Mills metrics on the vector bundle over a neighborhood of a singular fiber, one fiber of the fibration of the elliptic K3 surface over the base S^2. The second problem is to study on a compact balanced manifold the existence of a Hermitian metric which is balanced and Hermitian Yang-Mills with respect to itself. Such a metric can be viewed as a generalization of Kahler-Einstein metric on a compact Kahler manifold. The key step is to write down a desired flow which is a revision of the Hermitian Yang-Mills flow used by Donaldson.

考虑两个关于Hermitian Yang-Mills（HYM）度量的问题。第一个是理解在复环面或椭圆K3曲面上的某种稳定向量丛上的一类HYM度量的极限行为。方案是构造一类理想的Hermitian度量，再对这类度量与未知的HYM度量进行比较。对复环面的情形，已经获得了重要的结果，包括构造了一类理想的Hermitian度量，得到了零阶的下界估计和高阶估计。剩下的问题是估计零阶的上界。对椭圆K3曲面的情形，关键问题是当向量丛限制在奇异纤维的一个邻域上构造HYM度量，此处奇异纤维出现在椭圆K3曲面在S^2上的（奇异）纤维化。第二个问题是在紧平衡流形上研究既是平衡度量又是（相对于它自己的）HYM度量的存在性。这种度量可以看作是紧Kahler流形上的Kahler-Einstein度量的推广。问题的关键是要写下一个理想的流，它是对HYM流的某种修正。

复几何是一个重要的研究领域，它介于微分几何、多复变与偏微分方程之间。我们在这个研究领域获得了以下一系列成果。1. 研究了一类Hermitian-Yang-Mills度量的极限行为。2. 得到了紧近复流形上的延拓公式。3. 在紧Hermitian流形上定义了一个Hermitian曲率流并研究了它的正则性。4. 在紧凯勒流形上定义了一个新的deformed Hermitian-Yang-Mills流，研究了它的长时间解的存在性与收敛性，由此重新证明了Collins-Jacob-Yau的结果；也用这个流完全解决了二维情形时退化deformed Hermitian Yang-Mills方程解的存在性。5. 解决了Popa与Schnell于2014年提出的两个关于多典则丛高直像的Fujita猜想型的两个问题。