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Ancient Gauss Curvature Flows of Bounded Width

Beomjun Choi, Kyeongsu Choi, Dongjun Noh

TL;DR

This work develops ancient, non-self-similar solutions for curvature-driven flows in 3D: a pancake with flat sides under the Gauss curvature flow and sausage-type ancient solutions for the α-GCF with α>1/2. The authors establish a viscosity-solution framework, derive curvature-ratio evolution on rotationally symmetric profiles, and prove axisial minimality of curvature to enable barrier constructions. They construct the pancake via width-bounded approximations converging to a $C^{1,1}$ ancient solution, and obtain sausage solutions by both small-α barrier methods and large-α translator gluing, ensuring cylinder-like asymptotics and eventual extinction to a round point. These results illuminate flat-side persistence, width control, and asymptotic structures of ancient convex flows, contributing to the classification and behavior of non-self-similar ancient solutions in low dimensions.

Abstract

In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the $α$-Gauss curvature flow with $α>\frac{1}{2}$, asymptotic to a round cylinder.

Ancient Gauss Curvature Flows of Bounded Width

TL;DR

This work develops ancient, non-self-similar solutions for curvature-driven flows in 3D: a pancake with flat sides under the Gauss curvature flow and sausage-type ancient solutions for the α-GCF with α>1/2. The authors establish a viscosity-solution framework, derive curvature-ratio evolution on rotationally symmetric profiles, and prove axisial minimality of curvature to enable barrier constructions. They construct the pancake via width-bounded approximations converging to a ancient solution, and obtain sausage solutions by both small-α barrier methods and large-α translator gluing, ensuring cylinder-like asymptotics and eventual extinction to a round point. These results illuminate flat-side persistence, width control, and asymptotic structures of ancient convex flows, contributing to the classification and behavior of non-self-similar ancient solutions in low dimensions.

Abstract

In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the -Gauss curvature flow with , asymptotic to a round cylinder.

Paper Structure

This paper contains 10 sections, 23 theorems, 104 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Outline of the paper
  4. Preliminaries
  5. Curvature ratio evolution
  6. Ancient pancake with flat sides
  7. A priori estimate for speed
  8. Ancient sausage for flows by power of Gauss curvature
  9. Construction for small power
  10. Construction for large power

Key Result

Theorem 1.1

The Gauss curvature flow in $\mathbb{R}^3$ has an ancient pancake with flat sides, which is a viscosity solution of class $C^{1,1}$.

Figures (4)

  • Figure 1: Ancient pancake
  • Figure 2: Ancient sausage
  • Figure 3: Time slices of paperclip
  • Figure 4: Ancient sausage for $\alpha > 1$

Theorems & Definitions (49)

  • Theorem 1.1: ancient pancake with flat sides
  • Theorem 1.2: ancient sausage
  • Definition 2.1: viscosity solution, c.f. Section 8.2 andrews2000motion, Definition 2.6 choi2022convergence
  • Proposition 2.2: existence of unique solution, Theorem 15 andrews2000motion c.f. Theorem 2.7 choi2022convergence
  • Remark 2.3
  • Lemma 2.4: compactness
  • proof
  • Definition 2.5: displacement
  • Proposition 2.6: c.f. bourni2021collapsing
  • proof
  • ...and 39 more