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A classification result for eternal mean convex flows of finite total curvature type

Alexander Mramor

Abstract

In this article we partially classify the space of eternal mean convex flows in $\mathbb{R}^3$ of finite total curvature type, a condition implied by finite total curvature. In particular we show that topologically nonplanar ones must flow out of a catenoid in a natural sense.

A classification result for eternal mean convex flows of finite total curvature type

Abstract

In this article we partially classify the space of eternal mean convex flows in of finite total curvature type, a condition implied by finite total curvature. In particular we show that topologically nonplanar ones must flow out of a catenoid in a natural sense.
Paper Structure (11 sections, 9 theorems, 2 equations, 1 figure)

This paper contains 11 sections, 9 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of theorem
  4. Proof:
  5. Proof:
  6. Proof:
  7. Proof:
  8. Proof:
  9. Proof:
  10. Proof:
  11. Concluding remarks

Key Result

Theorem 1.1

Let $M_t \subset \Bbb R{$ R$}^3$, $t \in \Bbb R{$ R$}$, be a complete, embedded, connected eternal mean curvature flow of finite entropy such that: Then either $M_t$ is an annulus and flows out of a catenoid, in that as $t \to -\infty$$M_t$ converges to a catenoid from the outside, or is homeomorphic to $\Bbb R{$ R$}^2$.

Figures (1)

  • Figure 1: This figure illustrates a potential configuration of $M_t$ if it were to have more than 2 ends. Since the flow is mean convex and flows to either the empty set or union of planes, the top "neck" would have to collapse contradicting the smoothness of the flow.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Remark 3.1
  • ...and 1 more