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

• Applicant: Professor Miles Simon, Ph.D.
• Subject Area: Mathematics
• Term: from 2017 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 339362328

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity