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Foliation by bi-rotational self shrinkers in Euclidean space

This paper proves that the Euclidean space, excluding a compact set containing the origin, can be foliated by two families of bi-rotational asymptotically conical "trumpet" self shrinkers, two families of bi-rotational closed "disk" self shrinkers, and two generalized self-shrinking cylinders.

Original authors: Junyoung Park

Published 2026-08-12

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CC BY 4.0

Original authors: Junyoung Park

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a drop of water shrink on a leaf, or a soap bubble collapsing in on itself. In the world of mathematics, this shrinking process is called "mean curvature flow." It's a rule that says a shape will always try to shrink in the direction that reduces its surface area the fastest, kind of like a deflating balloon that wants to become as small as possible. But here's the tricky part: sometimes, as these shapes shrink, they don't just vanish smoothly. They can get pinched, develop sharp points, or even tear apart. These moments of chaos are called "singularities."

To understand what happens right before a shape tears itself apart, mathematicians use a special trick. They zoom in really, really close and slow down time, looking for "self-shrinkers." Think of a self-shrinker as a shape that shrinks perfectly in on itself, maintaining its exact same form the whole time, just getting smaller and smaller. It's like a snowflake that melts but somehow stays a perfect snowflake the entire time. These shapes are the "ghosts" of the singularities; they tell us what the universe looks like at the very moment things go wrong. For a long time, we knew about a few simple self-shrinkers, like cylinders (think of a rolling pin) and cones (like an ice cream cone). But the big question was: what about the weird, in-between shapes? Are there other hidden patterns that fill up the space around these known shapes?

This paper by Junyoung Park answers that question with a resounding "yes." The author proves that there are actually two new, distinct families of self-shrinking shapes that we can use to fill up almost all of the empty space in our universe, leaving only a small, messy cluster near the very center. He calls these new shapes "trumpets" and "disks."

The "trumpets" are fascinating. Imagine a shape that starts as a narrow tube and then flares out, getting wider and wider as it stretches out to infinity, looking a bit like a musical trumpet or a megaphone. The paper shows that there is a whole family of these, each with a slightly different flare, and they fit together perfectly like layers of an onion to fill the space between a cylinder and a cone.

The "disks" are the opposite. These are shapes that start wide and flat, like a pancake, and then curve inward to close up into a point, kind of like a dome or a bowl that is being squeezed shut. The author proves that there is also a family of these, and they fit together to fill the space inside the cylinders.

The most exciting part of the discovery is how these pieces fit together. Park proves that if you take these "trumpets," these "disks," the old "cylinders," and a special "minimal cone," you can arrange them so that they cover the entire universe, except for a small, compact ball right in the middle containing the origin. It's as if he found the missing puzzle pieces that allow us to map out the entire landscape of how shapes can shrink. He didn't just guess this; he built a rigorous mathematical proof, showing exactly how these shapes behave and how they slide past one another without overlapping. This gives mathematicians a complete map of the "far-field" behavior of these shrinking shapes, replacing a patchwork of guesses with a solid, continuous structure. It's a bit like realizing that the space between the trees in a forest isn't just random empty air, but is actually filled with a specific, repeating pattern of vines and leaves that we can now describe with perfect precision.

Technical Summary: Foliation by Bi-rotational Self Shrinkers in Euclidean Space

Problem Statement
The paper addresses the construction of foliations by self-shrinking solutions to the mean curvature flow (MCF) in Euclidean space $\mathbb{R}^{n+1}$ . Self-shrinkers are critical for understanding the formation of singularities in MCF, as they model the tangent flows at singular points. While rotationally symmetric self-shrinkers (such as cylinders and planes) and specific families like the "trumpets" (constructed by Kleene-Møller) are known, the existence of analogous foliations near generalized cylinders with bi-rotational symmetry ( $O(k) \times O(n-k+1)$ symmetry) remained an open question. Specifically, the author seeks to determine if the space surrounding these generalized cylinders can be foliated by families of self-shrinkers that interpolate between the cylinder, a minimal cone, and other asymptotic behaviors.

Methodology
The author employs a constructive approach based on ordinary differential equations (ODEs) derived from the self-shrinker equation under bi-rotational symmetry.

Reduction to ODEs: The hypersurfaces are parametrized by a profile curve $(x(s), y(s))$ in the first quadrant. The self-shrinker equation reduces to a second-order nonlinear ODE for a function $u(x)$ (where $y=u(x)$ ) or its inverse $v(y)$ .
Existence via Limiting Arguments:
- For "Trumpets": The author constructs a family of solutions $u_\sigma$ defined on $[x_s(\sigma), \infty)$ by solving an initial value problem with data at a large parameter $a$ and taking the limit as $a \to \infty$ . Uniform estimates (convexity, monotonicity, and asymptotic bounds) are established to ensure the convergence of a subsequence to a global solution.
- For "Disks": Similarly, a family of solutions $u_a$ defined on $[x_s(a), a]$ is constructed by solving an initial value problem for the inverse function $v(y)$ and taking the limit as the initial parameter $\epsilon \to 0$ .
Asymptotic Analysis: The paper derives precise asymptotic expansions for these solutions. For "trumpets," the solutions are shown to be asymptotically conical. For "disks," the solutions are analyzed in "tip" and "intermediate" regions to establish their behavior near the boundary and the axis.
Foliation Proof via Normal Variation: To prove that these families form a foliation (i.e., they do not intersect and cover the space), the author analyzes the normal variation $V = \nu \cdot \partial X / \partial \text{parameter}$ . By comparing $V$ against a barrier function $W = e^{|X|^2/8}$ (a strict supersolution to the linearized operator), the author proves that $V > 0$ for sufficiently large distances from the origin. This monotonicity ensures that the families sweep out the space without intersection.

Key Contributions and Results

Existence of New Families:
- Trumpets: Theorem 1.1 establishes the existence of a one-parameter family of bi-rotational, asymptotically conical self-shrinkers (diffeomorphic to $[0, \infty) \times S^{k-1} \times S^{n-k}$ ) that interpolate between a round cylinder and a minimal cone.
- Disks: Theorem 1.2 establishes the existence of a one-parameter family of bi-rotational self-shrinkers (diffeomorphic to $S^{k-1} \times D^{n-k+1}$ ) that close up, effectively filling the interior of the generalized cylinder.
Global Foliation Theorem:
- Theorem 1.3 (Theorem 5.1): The main result states that for $2 \le k \le n-1$ $2 \leq k \leq n - 1$ , there exists a compact set $K$ $K$ containing the origin such that the complement $\mathbb{R}^{n+1} \setminus K$ $R^{n + 1} ∖ K$ is foliated by:
  - The constructed "trumpets" and their reflections.
  - The constructed "disks" and their reflections.
  - Two generalized self-shrinking cylinders ( $S^{k-1}(\alpha_k) \times \mathbb{R}^{n-k+1}$ and $\mathbb{R}^k \times S^{n-k}(\beta_{n,k})$ ).
  - A minimal cone (the quadratic cone $|y| = \sigma_{n,k}|x|$ ).
Estimates and Regularity:
- The paper provides detailed $C^0$ and $C^1$ estimates for the profile functions, including bounds on the derivatives and asymptotic behavior as the parameter approaches the cone or cylinder limits.
- It is shown that the foliation satisfies key estimates analogous to those in previous works on rotationally symmetric flows, specifically regarding the behavior of unit normals near the cylinders.

Significance and Claims
The paper claims to generalize previous results on rotationally symmetric foliations (specifically those by Kleene-Møller and others) to the setting of bi-rotational symmetry. The primary significance lies in demonstrating that the geometric structure of self-shrinkers near generalized cylinders is rich enough to support a complete foliation of the exterior space.

The author notes that this construction provides an alternative tool for the analysis of ancient flows with bi-rotational symmetry. Specifically, Remark 1.4 states that these foliations can be used in place of "sub" self-shrinkers in the analysis of ancient ovals, potentially aiding in the study of unique asymptotics for such flows. The work does not propose new applications beyond this analytical utility in the study of mean curvature flow singularities and ancient solutions.

Technical Summary: Foliation by Bi-rotational Self Shrinkers in Euclidean Space

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