常纯量曲率与常边界平均曲率问题及其相关共形几何流

通过我们的一系列文章，我们致力于给出Zheng-Chao Han与Yan Yan Li 于1999年所提出的一个猜想的相对完整的肯定回答：若n(n≥3)维带边紧流形具有正的推广的Yamabe常数，则存在着一个共形度量使得它的纯量曲率为某个正常数同时它的边界平均曲率是任意实数。我们还将研究单位圆盘上的预定geodesic curvature 流以及单位球面上fractional scalar curvature 流的研究。

Through a series of our papers, we plan to give a relatively complete positive answer to a conjecture proposed by Zheng-Chao Han and Yan Yan Li (1999) in constant scalar curvature and constant boundary curvature problem: If an n-dimensional (n≥3) smooth compact manifold with boundary is of positive generalized Yamabe constant, there exists a conformal metric such that its scalar curvature equals a positive constant and its boundary mean curvature equals any real number. We also study some related conformal geometric flows, explicitly, prescribed geodesic curvature flow on the unit disk and fractional scalar curvature flow on the standard sphere.