Existence proofs for rotationally symmetric translating solutions to mean curvature flow

TL;DR

This paper demonstrates new methods to prove the existence of rotationally symmetric translating solutions to mean curvature flow by analyzing a singular ordinary differential equation, avoiding traditional PDE techniques.

Contribution

It introduces alternative approaches to establish existence results for these solutions through the study of the associated singular ODE, bypassing PDE methods.

Findings

Existence of rotationally symmetric translating solutions confirmed.

New methods based on singular ODE analysis developed.

Results provide alternative proofs to classical PDE-based proofs.

Abstract

There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.