Ancient Solutions to Null Mean Curvature Flow

• Applicant: Dr. Markus Wolff
• Subject Area: Mathematics
• Term: Funded in 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 571879798

The proposed project lies at the intersection of geometric analysis and general relativity. Geometric flows are one of the corner stones of modern geometric analysis and have a wide area of applications, including in mathematical physics. A solution of a geometric flow is called ancient if it exists for all past times. This usually implies a high level of rigidity which reflects back on the structure of the ambient space. Null mean curvature flow, a version of mean curvature flow along a null hypersurface, was first studied by Roesch and Scheuer and later by the applicant. In general relativity, a null hypersurface models the collective paths of all light that emanates off of a given light source, e.g., a star or galaxy. As information from far away celestial objects reaches us as radiation traveling along such null hypersurfaces, definitions of conserved quantities, such as mass and center of mass, are of particular physical relevance. The objective of the proposed project is to study the structure of ancient solutions to null mean curvature flow. More precisely, we want to obtain an independent characterization of ancient solutions of null mean curvature flow in the Minkowski lightcone, and to develop tools to construct ancient solutions of the flow in asymptotically flat lightcones. We further investigate the constructed solutions regarding definitions of center of mass and their applications toward geometric inequalities. A central tool in our analysis will be to develop of a suitable compactness theorem for the flow. Using this, one can consider parabolic rescalings and construct ancient solutions via a suitable family of large coordinate spheres. At null infinity, the mass should have a stabilizing effect on the level surfaces of the flow, implying a suitable rigidity and the desired properties of the solution. While geometric flows have proven a powerful tool and the study of ancient solutions of such flows is an active field of research, geometric flows along null hypersurfaces have only been studied in recent years. As definitions of conserved quantities in the null setting are likely related to real physical measurements, understanding their relation to the structure of geometric flows is highly relevant both from a mathematical and physical perspective. In addition, the structure of the level surfaces of the flow have significant advantages over other choices of foliations and could have potential applications to long standing problems such as the null Penrose inequality.

• DFG Programme: WBP Fellowship
• International Connection: Austria
• Host: Professor Michael Eichmair, Ph.D.