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Non-uniqueness of geodesic limits and a question of Grayson and Gage

This paper resolves a question posed by Grayson and Gage by constructing a smooth Riemannian metric on the 2-sphere and an immortal curve shortening flow that converges along different time sequences to a continuous family of distinct closed geodesics, thereby demonstrating that the limiting geodesic is not necessarily unique.

Original authors: Shrey Aryan, Tang-Kai Lee

Published 2026-08-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shrey Aryan, Tang-Kai Lee

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 piece of string floating in a magical, curved world. This string isn't just sitting there; it's alive with a desire to become as short and tight as possible. In the world of mathematics, this is called "curve shortening flow." Think of it like a rubber band that is constantly shrinking, but instead of snapping, it smoothly bends and twists to reduce its own length. If you put this shrinking string on a perfectly round ball (like a beach ball), mathematicians have known for a long time that the string will eventually settle down into a perfect circle around the middle, no matter where you started. It's a predictable, tidy ending.

But what if the world isn't a perfect ball? What if the surface is bumpy, lumpy, or shaped in a weird way? In 1990, two brilliant mathematicians named Grayson and Gage asked a big question: If you let this shrinking string run forever on a weirdly shaped surface, will it always settle down into just one specific shape (a "geodesic," which is the curved-surface version of a straight line)? Or could it get confused, spinning around and settling into different shapes depending on how you watch it? For decades, this was a mystery. Most people guessed that the string would eventually pick one final destination and stick to it, just like a ball rolling down a hill always stops at the bottom.

The Twist in the Tale

In this new paper, Shrey Aryan and Tang-Kai Lee say, "Not so fast!" They have proven that the answer to Grayson and Gage's question is no. They showed that you can build a very specific, slightly bumpy surface (on a sphere) where a shrinking string never picks just one final shape. Instead, it can get stuck in a loop, spiraling around and eventually settling into any shape from a whole family of different loops, depending on exactly when you stop watching it.

How They Did It: The Bumpy Ball and the Spiral Slide

To pull this off, the authors had to be like master architects. They couldn't just use a normal ball; they had to design a custom "bumpy" surface.

The Map of Lengths: First, they created a special map of the surface. Imagine the surface is covered in invisible hills and valleys. They designed it so that there is a whole ring of "perfect" loops (geodesics) that all have the exact same length. It's like a racetrack where every lane is the same distance, but they are all slightly different shapes.
The Spiral Slide: Next, they made the "gravity" of the shrinking string behave like a spiral slide. Usually, a shrinking string wants to roll down to the lowest point. But here, the authors engineered the surface so that the string doesn't roll straight down. Instead, it spirals around and around, getting closer and closer to that ring of perfect loops, but never quite stopping on just one.
The Magic of "Almost" Loops: They started with a string that was already very close to one of these perfect loops. As the string shrank, it didn't just snap into place. It followed the spiral slide. Because the slide is so tricky, the string could spiral around the ring of loops for a very long time.

The Big Discovery

The most exciting part is what happens at the end. Because the string is spiraling, if you check the string at time $T_1$ , it might look like it's settling into Loop A. If you check it at time $T_2$ (a little later), it might look like it's settling into Loop B. If you wait even longer, it could look like it's settling into Loop C.

The authors proved that for every loop in that special ring, there is a specific moment in time where the shrinking string looks like it has decided to become that exact loop. This means the string doesn't have a single, unique future. It has a whole family of possible futures, and which one you see depends entirely on when you look.

Why This Matters

This might sound like a game of "what if," but it's actually a huge deal for math. For a long time, mathematicians hoped that these shrinking processes were always predictable and unique. This paper proves that on certain surfaces, nature can be a bit more chaotic and indecisive. It shows that even in a world governed by strict rules, you can build a scenario where the final outcome isn't just one thing, but a whole collection of possibilities.

So, the next time you see a rubber band shrinking, remember: on a normal ball, it's boring and predictable. But on a specially designed, bumpy world, that rubber band could be the star of a show, spinning around and choosing a different costume every time you blink. The authors didn't just guess this; they built the math to prove it happens for real.

Technical Summary: Non-Uniqueness of Geodesic Limits and a Question of Grayson and Gage

Problem Statement
The paper addresses a fundamental question regarding the asymptotic behavior of the curve shortening flow (CSF) on closed surfaces. Grayson and Gage established that an immortal (existing for all $t \ge 0$ ) CSF of a simple closed curve on a closed surface converges subsequentially to a closed geodesic. However, they posed the question of whether this limiting geodesic is unique; that is, whether different sequences of times $t_i \to \infty$ must yield the same limiting geodesic. While uniqueness is known in specific settings—such as surfaces with negative curvature, the round sphere $(S^2, g_{\text{rd}})$ , and for real-analytic metrics (via Lojasiewicz–Simon arguments)—the general case for smooth metrics remained open. The authors aim to determine if non-uniqueness can occur for smooth Riemannian metrics on the sphere.

Methodology
The proof is constructive and proceeds in three main stages, combining geometric analysis, inverse function theorems, and dynamical systems:

Construction of Almost Geodesics (Section 2):
The authors construct a two-parameter family of "almost geodesics" $\{\Gamma_{\epsilon, v}\}$ on $S^2$ equipped with a conformal metric $g_\epsilon = e^{2\rho_\epsilon}g_{\text{rd}}$ . Utilizing the Nash–Moser inverse function theorem developed by Ambrozio–Marques–Neves [AMN25], they prescribe the length profile of these curves. Specifically, for a given smooth mean-zero function $W$ on the projective plane $P = \mathbb{RP}^2$ , they ensure that the length of $\Gamma_{\epsilon, v}$ is $L_\epsilon + \epsilon W(v)$ .
Crucially, they analyze the curvature vector $\vec{\kappa}_{\epsilon, \sigma}$ of these curves. They show that the component of the curvature tangent to the family of curves is exactly the negative gradient of the length profile, $-\nabla_{q_\epsilon}(\epsilon W)$ , while the transverse component is smaller by an additional factor of $\epsilon$ . This establishes that, to first order, the CSF moves through the family of almost geodesics according to the gradient flow of $W$ .
Dynamical Construction of the Potential (Section 3):
To generate non-unique limits, the authors construct a specific potential function $W$ on $\mathbb{RP}^2$ . They design $W$ such that its negative gradient flow contains a robust trajectory that spirals indefinitely around a prescribed embedded critical circle $Z \subset \mathbb{RP}^2$ .
The construction involves Fermi coordinates around $Z$ and a carefully crafted function where the speed of the flow tends to zero as it approaches $Z$ , but the logarithmic derivative of the speed along the trajectory can be made arbitrarily small. This ensures that the trajectory accumulates on the entire set $Z$ without converging to a single point, and that the flow is stable under small perturbations of the metric.
Mode Analysis and Stability (Section 4):
The authors analyze the CSF near a fixed almost geodesic by representing nearby curves as graphs over a reference curve. They decompose the graph function into "center modes" (the first two odd Fourier modes, corresponding to rotations of the great circle) and "stable modes" (higher odd Fourier modes).
Using a spectral gap argument, they prove that the stable modes decay exponentially faster than the center modes. Consequently, the long-term dynamics of the flow are governed entirely by the evolution of the center modes, which follow the spiraling gradient flow of $W$ constructed in Section 3. This allows them to control the full flow and ensure it accumulates on the specific family of geodesics indexed by $Z$ .

Key Results
The main result is stated in Theorem 1.1 and its detailed version Theorem 5.1:

There exists a smooth Riemannian metric $g$ on $S^2$ (which can be chosen arbitrarily close to the round metric) and an immortal simple closed curve shortening flow $\gamma(t)$ .
The set of subsequential limits of $\gamma(t)$ as $t \to \infty$ is not a single geodesic. Instead, it contains an entire one-parameter family of distinct, simple closed $g$ -geodesics $\{\Gamma_z\}_{z \in Z}$ , where $Z$ is an embedded circle in $\mathbb{RP}^2$ .
For every $z \in Z$ , there exists a sequence $t_i \to \infty$ such that $\gamma(t_i)$ converges smoothly to $\Gamma_z$ .
The geodesics in this family all have the same length.
For sufficiently large $t$ , the evolving curve $\gamma(t)$ intersects each limiting geodesic $\Gamma_z$ in exactly two points, demonstrating that the monotonicity of the set of intersecting geodesics (a tool used by Gage for the round metric) does not prevent non-uniqueness in this constructed setting.

Significance and Claims
The paper provides a negative answer to the question posed by Grayson and Gage, demonstrating that the uniqueness of the limiting geodesic for immortal curve shortening flows is not a universal property of smooth Riemannian metrics on $S^2$ .

The authors clarify that while uniqueness holds for the round metric (where the diameter of the set of intersecting geodesics shrinks to zero) and for negative curvature, the mechanism ensuring uniqueness fails for general smooth metrics.
The construction relies on the interplay between the geometry of the metric (via the Nash–Moser theorem) and the dynamics of the flow (via spectral gaps and gradient flow stability).
The paper does not claim to resolve the uniqueness question for real-analytic metrics (where a Lojasiewicz–Simon argument is expected to hold) or for higher-dimensional mean curvature flow, noting these as distinct open problems. The significance lies strictly in the existence of a smooth counterexample on the sphere.

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