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Existence proofs for rotationally symmetric translating solutions to mean curvature flow

Hakar Raji, Oliver C. Schnürer

TL;DR

This work proves the existence of rotationally symmetric translating solutions to mean curvature flow by reducing the graphical translator equation to a singular ODE for the radial derivative $\varphi=U'$. It develops five non-PDE techniques—weighted spaces, shooting, approximating problems, regularisation, and power series—to construct $\varphi$ on $[0,\infty)$ with the necessary regularity at the origin, yielding a globally defined translator $u(x)=\int_0^{|x|}\varphi(r)\,dr$ in $\mathbb{R}^n$ for $n\ge2$. Each method provides a distinct route to existence and regularity, enriching nonlinear analysis tools in geometric evolution problems. The results illuminate the structure of translational solutions and offer practical approaches for analyzing singular ODEs arising from geometric PDE reductions.

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.

Existence proofs for rotationally symmetric translating solutions to mean curvature flow

TL;DR

This work proves the existence of rotationally symmetric translating solutions to mean curvature flow by reducing the graphical translator equation to a singular ODE for the radial derivative . It develops five non-PDE techniques—weighted spaces, shooting, approximating problems, regularisation, and power series—to construct on with the necessary regularity at the origin, yielding a globally defined translator in for . Each method provides a distinct route to existence and regularity, enriching nonlinear analysis tools in geometric evolution problems. The results illuminate the structure of translational solutions and offer practical approaches for analyzing singular ODEs arising from geometric PDE reductions.

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.

Paper Structure

This paper contains 10 sections, 32 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Derivation of the ordinary differential equation
  3. Notation
  4. Derivation of the equation
  5. Existence away from from the origin
  6. Existence via weighted spaces
  7. Existence based on the shooting technique
  8. Existence based on approximating problems
  9. Existence based on a regularised equation
  10. Existence via power series

Key Result

Theorem 1.1

Let $n\ge2$. Then there exists a function ${\varphi}\colon[0,\infty)\to\mathbb{R}$ solving phi intro eq such that is of class $C^2$ and solves pde.

Figures (3)

  • Figure 1: Shooting method. We see a family of solutions with different inital values at $r=1$ and a solution that corresponds to the translator as a thick line.
  • Figure 2: Approximative solutions to \ref{['translat ode psi\n eq']}. In this figure, the black curve is the graph of $r\mapsto\frac{e^{-r}}{n-1}$, see Lemma \ref{['tr mcf ori up bd\n lem']}. All other curves in the picture solve \ref{['translat\n ode psi eq']}. The red curve corresponds to a translator which is smooth at the origin. The blue curves are $\psi_{k_0,-}$ and $\psi_{k_0,+}$ for ${\varepsilon}=\frac{1}{2}$ and $k_0=3,4,\ldots,12$. They pass through $(r,0)$ and $\left(r,\frac{e^{-r}}{n-1}\right)$, respectively, for $r={\varepsilon}\cdot k_0=1.5,2.0,\ldots,6.0$. In order to draw them, we use these points as initial values and solve \ref{['translat\n ode psi eq']} backwards up to $r=1.0$.
  • Figure 3: Approximative solutions to \ref{['translat ode psi eq']} as in Figure \ref{['shooting.fig']}, in logarithmic scaling.

Theorems & Definitions (73)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 63 more