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Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

José Torres Santaella

TL;DR

The paper develops a unified framework for rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows driven by an $\alpha$-homogeneous curvature function $\gamma$. It establishes precise bowl-type asymptotics for bowl-type translators, and constructs and classifies rotationally symmetric catenoidal translators $W_R$ under natural structural assumptions, including cases where $\gamma$ is signed or degenerate at the origin. Using barrier methods and an implicit-translator formulation, it proves rigidity and uniqueness results for graphical translators, notably showing that entire strictly convex translators asymptotic to bowl-type solitons coincide with them up to vertical translation. The results extend the classical mean curvature flow solitons to a broad nonlinear setting and provide tools for studying long-time behavior, desingularizations, and stability in $\gamma$-flows, with potential connections to $\Delta$-wing solitons and related geometric-analytic constructions.

Abstract

We study rotationally symmetric translators for fully nonlinear extrinsic geometric flows driven by a curvature function, and we establish the fine asymptotics of bowl-type evolutions and, when admissible, the construction and classification of catenoidal-type solutions, together with their asymptotic behavior. Under natural structural and convexity assumptions, we also prove rigidity and uniqueness results within appropriate classes of graphical translators of such curvature flows.

Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

TL;DR

The paper develops a unified framework for rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows driven by an -homogeneous curvature function . It establishes precise bowl-type asymptotics for bowl-type translators, and constructs and classifies rotationally symmetric catenoidal translators under natural structural assumptions, including cases where is signed or degenerate at the origin. Using barrier methods and an implicit-translator formulation, it proves rigidity and uniqueness results for graphical translators, notably showing that entire strictly convex translators asymptotic to bowl-type solitons coincide with them up to vertical translation. The results extend the classical mean curvature flow solitons to a broad nonlinear setting and provide tools for studying long-time behavior, desingularizations, and stability in -flows, with potential connections to -wing solitons and related geometric-analytic constructions.

Abstract

We study rotationally symmetric translators for fully nonlinear extrinsic geometric flows driven by a curvature function, and we establish the fine asymptotics of bowl-type evolutions and, when admissible, the construction and classification of catenoidal-type solutions, together with their asymptotic behavior. Under natural structural and convexity assumptions, we also prove rigidity and uniqueness results within appropriate classes of graphical translators of such curvature flows.
Paper Structure (13 sections, 31 theorems, 152 equations, 4 figures)

This paper contains 13 sections, 31 theorems, 152 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The class of $\gamma$-flows
  4. Translators
  5. Rotationally symmetric translators
  6. Implicit translator equations.
  7. Barrier methods
  8. Fine Asymptotic of Bowl-type Solutions
  9. The $1$-nondegenerate curvature function case.
  10. The $1$-degenerate curvature function case.
  11. Catenoidal translators
  12. Uniqueness of rotationally symmetric graphs and Applications
  13. Conclusions

Key Result

Proposition 2.7

Let $\gamma:\Gamma\to (0,\infty)$ be a positive $\alpha$-homogeneous curvature function. Then the level set can be described as the graph of $x=g(y,z)$, where $g_+(y,z)\in\mathcal{C}^1(U_+,V_+)$ is the implicit function of $\tilde{\gamma}(x,y)=z$ over open sets $U_+\subset{\mathbb R}^2$ and $V_+\subset{\mathbb R}$ given by:

Figures (4)

  • Figure 1: Catenoidal translator whose curvature function is continuous at the origin. Figure prepared by Ignacio McManus.
  • Figure 2: Catenoidal translator whose curvature function satisfies $g_-(0,-1)=0$ and $\partial_y g_-(0,-1)<0$. Figure prepared by Ignacio McManus.
  • Figure 3: $k=3$ the gorwth order is $r^{-1}$, $k=4$ the gorwth order is $-\ln(r)$, and the growth order is $-2r^{\frac{1}{2}}$
  • Figure 4: First contact point of $W_R$ and $\Sigma - t_1 \vec{e}_{n+1}$. Figure prepared by Ignacio McManus.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 57 more