Ricci flow on surfaces along the standard lightcone in the $$3+1$$-Minkowski spacetime

Abstract

Identifying any conformally round metric on the 2-sphere with a unique cross section of the standard lightcone in the $$3+1$$ 3 + 1 -Minkowski spacetime, we gain a new perspective on 2d-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch–Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from 2d-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we obtain a new proof of Hamilton’s classical result using only the maximum principle.