数量型曲率及其临界度量的几何、拓扑与分析的研究

数量型曲率主要包括数量曲率、Q-曲率以及共形变分黎曼不变量等黎曼几何中的基本曲率量；而其临界度量则是指对应的Einstein、Bach-flat以及更高阶的重要特殊度量。数量型曲率及其临界度量是黎曼几何的基本对象，一直以来在几何分析的研究中处于重要地位。本项目创造性地应用数学相对论的一些基本思想，研究数量型曲率及其临界度量的几何、拓扑与分析性质，其中包括：1. 几何课题：数量型曲率的体积比较定理的研究； 2. 拓扑课题：临界度量的间隙现象及其拓扑结论的研究； 3. 分析课题： 数量型曲率的Sobolev不等式的研究等三个部分。研究方法主要使用变分法结合几何偏微分方程的技术，通过分析和运用自然的几何结构，实现预期目标。本项目希望通过对数量型曲率及其临界度量的全方位研究，发现前所未知的新结果，从而增进对数量型曲率深刻性质的理解，并为解决几何学中一些长久以来悬而未决的猜想提供理论和工具上的支持。

Scalar-type curvatures include scalar curvature, Q-curvature and conformally variational Riemannian invariants etc. and all of them are fundamental curvatures in Riemannian geometry. Their critical metrics are referred to be those corresponding special metrics such as Einstein, Bach-flat and similar higher-order ones. Scalar-type curvatures and their critical metrics are fundamental objects in Riemannian geometry and they are also important research objects in the study of geometric analysis. By applying some basic ideas in Mathematical General Relativity creatively, this project mainly focus on researches on the geometric, topological and analytical properties of scalar-type curvature and their critical metrics. It includes three parts: 1. Geometric topics: study on the volume comparison of scalar-type curvatures; 2. Topological topics: study on gap phenomena of critical metrics and their topological consequences; 3. Analytical topics: study on Sobolev inequalities associated to scalar-type curvatures. The main methods will be variational methods together with geometric PDE technics. From analyzing the natural geometric structure, the presumed research purpose would be realized. We hope to find new results through systemically study on the scalar-type curvature and their critical metrics. This would provide us a deep understanding on scalar-type curvature and would give a support both theoretical and technical in solving some long-time standing conjectures in geometry.