K\”ahler曲面中全纯曲线的存在性问题是微分几何中的一个基本问题，这方面的问题近几十年来一直受到国内外数学家的广泛关注，并且已经得到很多结果，但是还有很多有意思的公开问题有待研究。我们将利用变分法，度量空间的几何流等新工具与研究微分几何和几何分析的经典工具黎曼几何，偏微分方程等结合起来研究相关的问题，包括：.（1）.考虑我们最近给出的全纯曲线的一个新的变分刻画的负梯度流的性质；.（2）.研究辛平均曲率流的第二类奇点的性质；.（3）.研究更多的与全纯曲线相关的泛函的临界点与稳定点的性质。.本项目旨在更加深入地了解全纯曲线的性质，并利用各种方法来考虑研究全纯曲线的存在性的可能性。本项目所涉及的问题是现在国际微分几何方向的热门话题之一，具有很高的学术研究价值。

Existence of holomorphic curves in a K\”ahler surface is a fundamental problem in differential geometry. It attracts much attention in the past several decades, and there are already many beautiful results on it. However, there are still many interesting open problems to be studied on this topic. We will combine some new tools, such as variational method, gradient flows in metric space, etc. with some classical ideas in studying differential goemetry and geometric analysis, for example, Riemannian geometry and partial differential equations together, to study related problems. It includes:.(1).The properties of the gradient flow of the variational problem we proposed for holomorphic curves recently..(2).The properties of Type II singularity of symplectic mean curvature flow..(3).The properties of critical points and stable points of more functionals related to holomorphic curves..This project aims at helping us to understand properties of holomorphic curves in-depth, and will use various ideas to attach the possibility of the center of the problem-existence of holomorphic curves. The problems mentioned in this project have high academic research value, as they are related to one of the hottest topics in differential geometry.