The PDE-ODI principle and cylindrical mean curvature flows

TL;DR

This paper introduces the PDE--ODI principle, a new method converting parabolic PDEs into ODE systems, enabling high-order asymptotic analysis of cylindrical mean curvature flows and unifying classical results with simpler proofs.

Contribution

The paper presents the PDE--ODI principle, a novel approach that simplifies analysis of ancient solutions and singularities in mean curvature flow, extending and unifying previous results.

Findings

Proves uniqueness of the bowl soliton times a Euclidean factor for cylindrical flows.

Provides complete asymptotic expansions to arbitrary polynomial order.

Recovers classical results on tangent flow uniqueness and cylinder rigidity with new proofs.

Abstract

We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the \emph{PDE--ODI principle}, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the {\L}ojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.