[Paper Review] A new version of Brakke's local regularity theorem

Ananda Lahiri|arXiv (Cornell University)|Jan 25, 2016

Geometric Analysis and Curvature Flows20 references3 citations

TL;DR

This paper presents a new version of Brakke's local regularity theorem for integral Brakke flows in Euclidean space, extending the time interval of smooth graphical regularity up to the final time $ t_2 $, under assumptions of small initial height, measure bounds, and non-vanishing mass. The key advance is proving that flows initially locally graphical with small gradient remain graphical and smooth up to $ t_2 $, generalizing both Brakke’s and White’s regularity theorems to the weak setting of Brakke flows.

ABSTRACT

Consider an integral Brakke flow $(μ_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $μ_{0}$ has small height, its measure does not deviate too much from that of a plane and if $μ_{T}$ is non-empty, then Brakke's local regularity theorem yields that $(μ_t)$ is actually smooth and graphical inside a smaller ball for times $t\in (C,T-C)$ for some constant $C$. Here we extend this result to times $t\in (C,T)$. The main idea is to prove that a Brakke flow that is initially locally graphical with small gradient will remain graphical for some time. Moreover we use the new local regularity theorem to generalise White's regularity theorem to Brakke flows.

Motivation & Objective

• To extend Brakke's local regularity theorem to include regularity up to the final time $ t_2 $, rather than only up to $ t_2 - C $, in the context of integral Brakke flows.
• To establish that a Brakke flow initially locally graphical with small gradient remains graphical and smooth for a positive time interval including $ t_2 $, under measure and height constraints.
• To generalize White’s local regularity theorem to Brakke flows by showing that Gaussian density ratios close to one imply curvature estimates and graphical regularity.
• To provide a rigorous framework for regularity in weak mean curvature flow by refining the use of monotonicity formulas and measure-theoretic estimates.

Proposed method

• Use of Huisken’s monotonicity formula and its localization via Ecker’s cutoff functions to control measure decay and curvature behavior.
• Introduction of a new time-dependent graphical representation $ u(t, ullet) $ with controlled gradient $ \sup |Du| \leq \alpha_0^{-1} \rho^{-2}(t - t_1) $ to ensure smoothness.
• Application of a refined height bound and measure comparison with a plane to control deviation from flatness in the initial slab.
• Establishment of a new local regularity condition based on Gaussian density ratios close to one, analogous to White’s theorem, but extended to Brakke flows.
• Use of iterative subsequential convergence arguments and compactness to construct a limiting Brakke flow on nested domains, ensuring consistency across scales.
• Leveraging Ilmanen’s and Ecker’s work on monotonicity and curvature estimates to derive uniform bounds under small initial data conditions.

Experimental results

Research questions

• RQ1 Can Brakke’s local regularity theorem be extended to guarantee smooth graphical regularity up to the final time $ t_2 $, rather than only up to $ t_2 - C $?
• RQ2 Under what conditions does a Brakke flow that is initially locally graphical with small gradient remain graphical and smooth for a positive time interval including $ t_2 $?
• RQ3 Can White’s curvature estimate based on Gaussian density ratios be generalized to the setting of integral Brakke flows?
• RQ4 How can measure-theoretic and geometric constraints—such as small height, small measure deviation from a plane, and non-vanishing mass—be used to ensure long-time regularity?
• RQ5 What is the optimal time interval over which graphical regularity persists for a Brakke flow under small initial data and measure bounds?

Key findings

• A new local regularity theorem is established that guarantees smooth graphical regularity for Brakke flows up to time $ t_2 $, provided the initial measure lies in a narrow slab of height $ \gamma\rho $, with $ \gamma \in [0, \gamma_0] $, and the measure at $ t_1 $ is bounded by $ (2 - \lambda)\omega_n \rho^n $.
• The time interval of regularity is extended to $ I = (t_1 + \gamma^{\alpha_0}\rho^2, t_2) $, where $ \alpha_0 \in (0,1) $, showing that regularity holds up to the final time $ t_2 $, not just before it.
• A graphical representation $ u \in C^\infty(I \times \mathbf{B}^n(\hat{a}, \gamma_0\rho), \mathbb{R}^k) $ exists such that $ \mu_t \mathop{\vrule height=5.59721pt,depth=0.0pt,width=0.51663pt\vrule height=0.4736pt,depth=0.0pt,width=4.73611pt} \mathbf{C}(a, \gamma_0\rho, \rho) = \mathscr{H}^n \mathop{\vrule height=5.59721pt,depth=0.0pt,width=0.51663pt\vrule height=0.4736pt,depth=0.0pt,width=4.73611pt} \mathrm{graph}(u(t, \cdot)) $, ensuring smoothness and graphicality.
• The gradient of the graph is uniformly controlled: $ \sup |Du(t, \cdot)| \leq \alpha_0^{-1} \rho^{-2}(t - t_1) $, which ensures the flow remains graphical and smooth for all $ t \in I $.
• A new White-type regularity theorem is proven: if the Gaussian density ratio is close to one, then the Brakke flow becomes locally graphical and smooth, generalizing White’s result to the weak setting.
• The proof relies on refined measure estimates, monotonicity formulas, and iterative compactness arguments to construct a consistent limiting Brakke flow across nested domains, ensuring regularity propagation.