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Killing Mean Curvature Solitons from Riemannian Submersions

Diego Artacho, Marie-Amélie Lawn, Miguel Ortega

TL;DR

This work introduces a general method for constructing mean curvature flow solitons on manifolds with a nowhere-vanishing Killing field by exploiting a warped-product and Riemannian submersion framework to convert a PDE into an ODE. Specifically, for ambient spaces $N\cong M\times_{\varphi} I$, graph solitons correspond to solving $\mathrm{div}\left(\frac{\nabla u}{W}\right)=\frac{1}{W}-\frac{g_M(\nabla\varphi,\nabla u)}{2W\varphi}$ with $W=\sqrt{g_M(\nabla u,\nabla u)+1/\varphi}$, and $K=\partial_r$ encodes the soliton flow. When the setting further factors through a base $J$ via a submersion $\pi:(M,g_M)\to(J,\alpha(s)ds^2)$ with $\varphi=\hat{\varphi}\circ\pi$ and fibres of constant mean curvature $h$, the PDE reduces to the ODE $f''=\left(\alpha+\hat{\varphi}(f')^2\right)\left(1-\frac{\hat{\varphi}'f'}{2\hat{\varphi}\alpha}-\frac{f'h}{\sqrt{\alpha}}\right)+\frac{f'}{2}\left(\log\left(\frac{\alpha}{\hat{\varphi}}\right)\right)'$. The authors apply this to hyperbolic space, producing a one-parameter family of complete solitons (rotators) in $\mathbb{H}^{n+1}$, including bowl and wing-like variants, thereby expanding known examples and illustrating the method's geometric versatility.

Abstract

We present a new general construction of examples of mean curvature solitons on manifolds admitting a nowhere-vanishing Killing vector field. Using Riemannian submersion techniques, we reduce the problem from a PDE to an ODE. As an application, we obtain new examples of rotators in hyperbolic space.

Killing Mean Curvature Solitons from Riemannian Submersions

TL;DR

This work introduces a general method for constructing mean curvature flow solitons on manifolds with a nowhere-vanishing Killing field by exploiting a warped-product and Riemannian submersion framework to convert a PDE into an ODE. Specifically, for ambient spaces , graph solitons correspond to solving with , and encodes the soliton flow. When the setting further factors through a base via a submersion with and fibres of constant mean curvature , the PDE reduces to the ODE . The authors apply this to hyperbolic space, producing a one-parameter family of complete solitons (rotators) in , including bowl and wing-like variants, thereby expanding known examples and illustrating the method's geometric versatility.

Abstract

We present a new general construction of examples of mean curvature solitons on manifolds admitting a nowhere-vanishing Killing vector field. Using Riemannian submersion techniques, we reduce the problem from a PDE to an ODE. As an application, we obtain new examples of rotators in hyperbolic space.
Paper Structure (6 sections, 8 theorems, 79 equations, 4 figures)

This paper contains 6 sections, 8 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The PDE
  3. The ODE
  4. An Application to the Hyperbolic Space
  5. Some Technical Lemmas
  6. Calculation of the mean curvature \ref{['eq:mean_curvature']}

Key Result

Theorem 1

In our previous setting, the graph of $u=f\circ \pi$ defines a $\mathcal{G}$-soliton on $N$ if and only if $f$ satisfies the ordinary differential equation:

Figures (4)

  • Figure 1: Intersection of our new $\mathcal{G}$-soliton on $\mathbb{H}^3$ with the plane $x_1 = 1$.
  • Figure 2: A new complete $\mathcal{G}$-soliton on $\mathbb{H}^3$.
  • Figure 3: Intersection of a wing-like $\mathcal{G}$-soliton with the plane $x_1 = 1$.
  • Figure 4: A complete wing-like $\mathcal{G}$-soliton on $\mathbb{H}^3$.

Theorems & Definitions (21)

  • Theorem : see Theorem \ref{['thm:ode']}
  • Theorem 1
  • proof
  • Remark 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • proof
  • Remark 7
  • ...and 11 more