[Paper Review] Closed hypersurfaces of low entropy in $\mathbb{R}^4$ are isotopically trivial

Jacob Bernstein, Lu Wang|arXiv (Cornell University)|Mar 30, 2020

Geometric Analysis and Curvature Flows25 references4 citations

TL;DR

This paper proves that any closed, connected hypersurface in $ℝ^4$ with entropy less than or equal to that of the round cylinder $σ^2 \times \mathbb{R}$ is smoothly isotopic to the standard 3-sphere $σ^3$. Using mean curvature flow and tangent flow analysis, the authors establish that low-entropy hypersurfaces in $ℝ^4$ cannot develop nontrivial singularities, forcing them to be topologically trivial via isotopy.

ABSTRACT

We show that any closed connected hypersurface in $\mathbb{R}^4$ with entropy less than or equal to that of the round cylinder is smoothly isotopic to the standard three-sphere.

Motivation & Objective

• To resolve the Schoenflies problem in $ℝ^4$ for hypersurfaces with low entropy, a geometric analog of topological triviality.
• To establish that closed connected hypersurfaces in $ℝ^4$ with entropy $\leq \lambda[\mathbb{S}^2 \times \mathbb{R}]$ are smoothly isotopic to $\mathbb{S}^3$, extending known results in lower dimensions.
• To provide a sharp topological classification of low-entropy hypersurfaces in $ℝ^4$ using mean curvature flow and tangent flow analysis.
• To confirm that the only possible singularity models for such flows are isotopic to $\mathbb{S}^3$, ruling out nontrivial minimal cones or self-shrinkers.

Proposed method

• Utilizes the entropy functional $\lambda[\Sigma] = \sup_{\mathbf{y}, \rho} F[\rho\Sigma + \mathbf{y}]$, where $F$ is the Gaussian surface area, to measure geometric complexity.
• Applies mean curvature flow with a matching motion $\mathcal{K} = \{\mu_t\}_{t \geq 0}$ to evolve the hypersurface, ensuring non-fattening and regularity via short-time existence.
• Analyzes tangent flows at singularities using the White regularity theorem and Brakke’s regularity theory to rule out non-compact or non-isotopic models.
• Employs the concept of $(R_0, C_0)$-regular almost a.c.-isotopies to control the structure of tangent flows and ensure isotopy to $\mathbb{S}^3$.
• Applies Huisken’s monotonicity formula to show entropy decreases along the flow, allowing reduction to a non-fattening initial hypersurface with $\lambda[\Sigma] < \Lambda - \epsilon_0$.
• Uses the fact that all tangent flows at extinction time are isotopic to $\mathbb{S}^3$, implying the original hypersurface is smoothly isotopic to $\mathbb{S}^3$.

Experimental results

Research questions

• RQ1 Can closed hypersurfaces in $\mathbb{R}^4$ with entropy bounded by that of $\mathbb{S}^2 \times \mathbb{R}$ be topologically trivial via smooth isotopy?
• RQ2 Do low-entropy hypersurfaces in $\mathbb{R}^4$ avoid forming nontrivial singularities under mean curvature flow?
• RQ3 Is the only possible singularity model for such flows isotopic to $\mathbb{S}^3$?
• RQ4 Can the conditional result on low-entropy hypersurfaces in higher dimensions be strengthened to a full classification in $\mathbb{R}^4$?
• RQ5 Does the absence of non-flat self-shrinkers and minimal cones with entropy below $\lambda[\mathbb{S}^2 \times \mathbb{R}]$ imply isotopy to $\mathbb{S}^3$?

Key findings

• Any closed connected hypersurface $\Sigma \subset \mathbb{R}^4$ with $\lambda[\Sigma] \leq \lambda[\mathbb{S}^2 \times \mathbb{R}]$ is smoothly isotopic to $\mathbb{S}^3$.
• The proof establishes that all tangent flows at singularities are either compact or $(R_0, C_0)$-regular almost a.c.-isotopies, ensuring isotopy to $\mathbb{S}^3$.
• The extinction time of the mean curvature flow for such hypersurfaces occurs at a single point, and the singularity model is isotopic to $\mathbb{S}^3$.
• The entropy of $\Sigma$ decreases under mean curvature flow, allowing reduction to a non-fattening initial hypersurface with strictly less entropy.
• The absence of non-flat elements in $\mathcal{S}_3^*(\Lambda)$ and $\mathcal{RMC}_k^*(\Lambda)$ for $k \leq 3$ is essential to the classification.
• The result confirms a geometric version of the Schoenflies problem in $\mathbb{R}^4$ for low-entropy hypersurfaces, extending prior results in $\mathbb{R}^3$.