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

TL;DR

This paper links 2D Ricci flow on spheres to null mean curvature flow in Minkowski spacetime, providing new insights and proofs for singularity models and classical results through geometric and analytical methods.

Contribution

It establishes an equivalence between Ricci flow on spheres and null mean curvature flow in Minkowski space, enabling new proofs and classifications.

Findings

Equivalence between 2D Ricci flow and null mean curvature flow.

Classification of singularity models for null mean curvature flow.

New proof of Hamilton's classical Ricci flow result using maximum principle.

Abstract

Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-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.