Solutions to Ricci flow whose scalar curvature is bounded in L^p (II)

当p&gt；=n/2时，我们计划研究n维Ricci流的有限时间奇点的结构，如果标量曲率保持全局或局部有界。在项目第一个资助期产生的预印本中，申请人和博士后刘家伟博士推广了申请人早期工作的结果。如果解定义在闭Kähler 2n维(复n维)流形或实四维闭流形上，则：a)空间中Ricci曲率的积分L^q范数在时空中是有界的，其中q和S是依赖于n和p的显式常数.b)黎曼曲率的L^2范数是有界的.我们进一步证明了a)和b)的估计可以在Kähler环境下局部化.在四个维度上，我们计划使用申请人早期工作的这些积分估计和集中紧性方法，在适当的情况下，Bamler和Bamler-Zhang的方法，证明存在局部(或在真实情况下是全局的)极限，T是奇异时间。在Kähler局部环境中，我们计划在一般的维度上进一步发展这些方法，并证明存在局部度量空间极限t-&gt；T，T是奇点时间，并研究了它的结构，我们计划在四维空间中建立一个局部的或分叉的Ricci流，在高维空间中建立一个局部锥形Ricci流或其他局部弱Ricci流，以使流在奇点时间之后继续。

We plan to investigate the structure of finite time singularities of the Ricci flow in n dimensions if the scalar curvature remains globally or locally bounded in L^p for p>= n/2. In a preprint resulting from the first funding period of the project, the applicant and the Post-Doc Dr. Jiawei Liu, generalised results of earlier work of the applicant. We show in this preprint that if the solution is defined on a closed Kähler 2n-dimensional ( complex n-dimensional ) manifold, or a real four dimensional closed manifold, then:a) the integral L^q norm of of the Ricci curvature in space respectively L^s norm in space-time is bounded where q and s are explicit constants depending on n and p. b) the L^2 norm of the Riemannian curvature is bounded.We further show that the estimates of a) and b) can be localised in the Kähler setting. In four dimensions, we plan to use these integral estimates and concentration compactness methods of earlier work of the applicant, and where appropriate those of Bamler and Bamler-Zhang, to show that there is a local (or global in the real case) orbifold limit as t --> T, T being the singular time.In the Kähler local setting we plan to develop further these methods in general dimensions, and prove that there is a local metric space limit as t-->T, T being the singular time, and to study the structure thereof.We plan to develop a local orbifold Ricci flow in four dimensions and a local conical Ricci flow or other local weak Ricci flow in higher dimensions in order to continue the flow past the singular time.