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A gradient estimate for the linearized translator equation

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

Abstract

In this paper, we develop some analytic foundations for the linearized translator equation in $\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in $\mathbb{R}^4$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_v$, namely for the derivative of the variation field $W$ in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling $Y_v$, namely the derivative of the profile function $Y$ in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for $W_τ$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_τ$ in the tip region.

A gradient estimate for the linearized translator equation

Abstract

In this paper, we develop some analytic foundations for the linearized translator equation in , i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for , namely for the derivative of the variation field in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling , namely the derivative of the profile function in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling in the tip region.

Paper Structure

This paper contains 4 sections, 14 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Precise profile estimates for the bowl soliton
  3. An enhanced quadratic concavity estimate for the oval-bowls
  4. The gradient estimate for the linearized translator equation

Key Result

Theorem 1.1

Let $\mu\geq 0$. Suppose that $A<\infty$ is a constant such that and suppose that for all $\tau\in [-\log (h),\tau_0]$ we have Then, there is some $C=C(\mu,\tau_0,\ell,\theta)<\infty$, such that for all $\tau\in [-\log (h)^{1/2},\tau_0]$ we get

Figures (3)

  • Figure 1: The oval-bowls $\{M_{\kappa}\}_{\kappa\in(0,1/3)}$ are noncollapsed translators in $\mathbb{R}^4$, whose level sets look like 2d-ovals in $\mathbb{R}^3$. The principal curvatures at the origin are $(\kappa,\tfrac{1-\kappa}{2},\tfrac{1-\kappa}{2})$.
  • Figure 2: The function $K$ (blue) and the lower and upper bounds (orange and green).
  • Figure 3: The function $\delta$ (blue) is strictly larger than $0$ (orange).

Theorems & Definitions (26)

  • Theorem 1.1: inner-outer estimate, CHH_lin_trans
  • Theorem 1.2: gradient estimate
  • Proposition 2.1: quantitative bounds
  • proof
  • Corollary 2.2: coefficient estimate
  • proof
  • Lemma 3.1: third derivative decay
  • proof
  • Proposition 3.2: enhanced third derivative estimate
  • proof
  • ...and 16 more