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Monotonicity of the modulus under curve shortening flow

The paper demonstrates that for two disjoint nested embedded closed curves evolving under curve shortening flow, the modulus of the annulus they enclose increases monotonically over time, a result that extends to ambient surfaces with a lower curvature bound.

Original authors: Arjun Sobnack, Peter M. Topping

Published 2026-04-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Arjun Sobnack, Peter M. Topping

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 have two rubber bands floating in a pond. One is inside the other, so they form a ring-shaped space (an annulus) between them. Now, imagine these rubber bands are magical: they are constantly trying to shrink themselves as fast as possible to become as small as they can. This shrinking process is called Curve Shortening Flow.

In this paper, mathematicians Arjun Sobnack and Peter Topping ask a very specific question: As these two rubber bands shrink, does the "shape" of the space between them change in a predictable way?

The Big Discovery: The "Stretchy" Space

The authors discovered a beautiful rule: As the rubber bands shrink, the space between them gets "stretched out" or "thinner" in a very specific mathematical sense.

To understand this, we need to talk about something called the Modulus.

The Analogy: The Rubber Hose

Think of the space between your two rubber bands as a long, flexible garden hose.

If the hose is short and fat, it's easy to walk through it.
If the hose is very long and skinny, it's harder to traverse; it feels "stretched."

In mathematics, the Modulus is a number that measures exactly how "long and skinny" this hose is.

Low Modulus: A short, fat ring (like a thick bracelet).
High Modulus: A long, thin ring (like a long tunnel).

The Main Result: The paper proves that as the rubber bands shrink under their own tension, the Modulus always increases. The space between them is constantly getting "longer and skinnier." It never gets shorter or fatter.

Why is this surprising?

Usually, when things shrink, they just get smaller. But here, the two rubber bands are shrinking independently. They don't talk to each other. Yet, because they are both shrinking in the same way, they act like a team. They push the space between them to stretch out.

The authors show that this stretching happens at a steady, predictable rate, especially if the surface they are on (the "pond") isn't too curvy. If the pond is perfectly flat (like a sheet of paper), the space stretches out perfectly. If the pond is curved (like a sphere), the stretching still happens, but the curvature of the pond adds a little extra "push" or "drag" to the process.

The "Grim Reaper" Connection

The paper starts with a fun example called the Grim Reaper. Imagine a curve that looks like a wave. If you slide this wave sideways at a constant speed, it looks like it's flowing. The authors realized that the space "under" this wave is actually a special kind of infinite tunnel. This helped them understand how to measure the "shape" of the space between any two shrinking loops.

Why should you care? (The "Why it matters" part)

You might wonder, "Who cares about shrinking rubber bands?"

This isn't just about rubber bands. This math is a tool for understanding how shapes behave when they change over time.

In Physics: It helps model how interfaces between materials move (like oil and water separating).
In Geometry: It helps mathematicians prove that shapes don't get "weird" or "broken" when they evolve.
The "Barrier" Effect: The paper suggests that if you have a "nice" inner loop (like a perfect circle shrinking) and a "wild" outer loop (like a crumpled piece of paper), the nice inner loop acts as a barrier. It forces the wild outer loop to smooth out and behave nicely as time goes on. The fact that the space between them stretches predictably is the key to proving this.

Summary in Plain English

The Setup: Two loops, one inside the other, are shrinking to become points.
The Rule: The space between them doesn't just shrink; it transforms. It becomes "longer and skinnier" (its Modulus increases).
The Guarantee: This stretching is guaranteed to happen, no matter how weird the loops look, as long as they don't crash into each other.
The Takeaway: Even when things are shrinking and changing chaotically, there are hidden, orderly rules governing the space between them. The universe prefers to stretch the space between shrinking objects rather than letting it crumple.

It's like watching two people walk toward each other in a hallway; as they get closer, the hallway between them doesn't just get shorter—it feels like the hallway is stretching out to accommodate their movement in a very specific, mathematical way.

1. Problem Statement

The paper investigates the geometric evolution of the region bounded by two disjoint, nested, embedded closed curves in a Riemannian surface when both curves evolve simultaneously under the Curve Shortening Flow (CSF).

Context: In the standard CSF, a single curve evolves in the direction of its geodesic curvature vector ( $\partial_t \gamma = \vec{\kappa}$ ). It is well-known that a single loop in $\mathbb{R}^2$ shrinks to a point in finite time.
The Setup: Consider two such loops, an inner loop $\gamma_0$ and an outer loop $\gamma_1$ , which are initially disjoint and nested. By the avoidance principle (maximum principle), they remain disjoint as they evolve, thereby enclosing an annular region $A_t$ .
The Question: How does the conformal geometry of this evolving annulus change over time? Specifically, the authors analyze the modulus ( $h$ ) of the annulus, a conformal invariant that characterizes the "width" of the annulus relative to its boundary lengths.

2. Methodology

The authors employ a combination of conformal geometry, variational calculus, and geometric analysis (specifically the Bochner technique).

A. Conformal Characterization of the Modulus

The paper utilizes the Uniformization Theorem, which states that any annulus is biholomorphic to a cylinder $(0, h) \times S^1$ . The modulus $h$ is defined via the Dirichlet energy (or capacity) of the annulus.

Let $u_t: A_t \to [0, 1]$ be the unique harmonic function on the annulus taking value 0 on the inner boundary and 1 on the outer boundary.
The energy is defined as $E(u_t) = \frac{1}{2} \int_{A_t} |\nabla u_t|^2 d\mu$ .
The modulus is related to this energy by the formula:
$h = \frac{1}{2 E(u_t)}$
(Note: The paper defines $S^1$ to have length 1).

B. Variation of Energy

To determine how $h$ changes, the authors compute the time derivative of the energy $E(u_t)$ as the domain $A_t$ evolves.

Domain Variation: They consider the boundary $\partial A_t$ moving with normal speed $\phi$ .
Harmonic Function Variation: They derive the evolution of the harmonic function $u_t$ itself. Since $u_t$ is constant on the boundaries, the time derivative of $u$ at the boundary is determined by the normal speed $\phi$ and the normal derivative of $u$ .
Energy Evolution Formula: Using the divergence theorem and properties of harmonic functions, they derive the rate of change of energy:
$\frac{d}{dt} E(u_t) = -\frac{1}{2} \int_{\partial A_t} |\nabla u|^2 \phi \, ds$

C. Application to Curve Shortening Flow

The specific case of interest is when the boundary curves evolve under CSF.

The normal speed is $\phi = -\kappa$ , where $\kappa$ is the geodesic curvature.
The authors establish a key identity relating the normal derivative of the gradient magnitude to the curvature:
$\frac{1}{2} \frac{\partial}{\partial \nu} |\nabla u|^2 = -\kappa |\nabla u|^2$
Substituting this into the energy evolution formula allows them to convert the boundary integral into a volume integral over the annulus $A_t$ .

D. Bochner Technique

The final step involves applying the Bochner formula to the harmonic function $u$ .

The Bochner formula relates the Laplacian of the gradient squared to the Hessian of the function and the Gauss curvature $K$ of the ambient surface:
$\Delta |\nabla u|^2 = 2|\text{Hess}(u)|^2 + 2K|\nabla u|^2$
By integrating this over the annulus and using the divergence theorem, they relate the change in energy to the curvature of the surface and the Hessian of the harmonic map.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

The central result provides a lower bound on the rate of change of the modulus $h$ for an annulus evolving under CSF in a Riemannian surface $M$ with Gauss curvature bounded below by $K_0$ :
$\frac{d}{dt} \log h \geq K_0$

Specific Implications:

Euclidean Plane ( $K_0 = 0$ ): If the ambient space is $\mathbb{R}^2$ $R^{2}$ , then $\frac{d}{dt} \log h \geq 0$ $\frac{d}{d t} lo g h \geq 0$ . Since the inequality is strict unless the geometry is trivial, the modulus is strictly increasing.
- Theorem 1.1: For two disjoint nested loops in $\mathbb{R}^2$ , the modulus of the enclosed annulus is strictly increasing in time.
General Surfaces: If the surface has positive curvature lower bound, the modulus grows at least exponentially. If the curvature is negative, the growth is slowed but still controlled by the bound.
Rigidity Case: Equality holds ( $\frac{d}{dt} \log h = K_0$ ) if and only if $K_0 = 0$ and the annulus is isometric to a flat cylinder $(0, h) \times S^1$ . This implies the curves are concentric circles in a flat plane.

4. Significance and Motivation

Regularity Estimates for CSF: The primary motivation for this work is to establish geometric constraints that can lead to regularity estimates for solutions of the Curve Shortening Flow. The authors propose a framework where the behavior of a "wild" outer curve ( $\gamma_1$ ) can be controlled by a "regular" inner curve ( $\gamma_0$ ) if they enclose a small area $A_0$ .
Interaction of Curves: The results highlight that while curves evolve independently, their interaction as barriers creates a monotonic geometric quantity (the modulus). This monotonicity is a powerful tool for proving that singularities or irregularities in the outer curve cannot develop arbitrarily fast if constrained by the inner curve.
Connection to Ricci Flow: The paper notes a bridge between this theory and Ricci flow, specifically referencing $L^1-L^\infty$ smoothing estimates. The monotonicity of the modulus serves as a conformal geometric analog to entropy monotonicity in Ricci flow.
Technical Advancement: The proof simplifies previous approaches that relied on direct biholomorphic mappings. Instead, it uses a variational approach based on harmonic functions and the Bochner formula, which is more robust and applicable to general Riemannian surfaces.

Summary Conclusion

Sobnack and Topping demonstrate that the conformal modulus of an annulus formed by two nested curves evolving under Curve Shortening Flow is a monotonically increasing quantity (in the Euclidean case). This result is derived by analyzing the evolution of the Dirichlet energy of the harmonic function defining the annulus, utilizing the Bochner formula to link this evolution to the ambient curvature. This monotonicity provides a crucial geometric tool for understanding the long-term behavior and regularity of non-graphical Curve Shortening Flow solutions.