Quadratic growth of geodesics on the two-sphere

This paper proves that for any reversible Finsler metric on the two-sphere, the number of prime closed geodesics grows quadratically with respect to length, utilizing improved versions of Franks' theorem and cylindrical contact homology.

The Gist

Here is an explanation of Bernhard Albach's paper, "Quadratic Growth of Geodesics on the Two-Sphere," translated into simple, everyday language with creative analogies.

The Big Picture: Counting Loops on a Ball

Imagine you have a perfect, round ball (a sphere). Now, imagine you wrap a rubber band around it. If you let the rubber band snap tight, it forms a geodesic—the shortest possible path between two points on that surface. On a sphere, these are great circles (like the equator).

But what if the ball isn't perfectly round? What if it's slightly squashed, or has bumps, or is made of a weird, stretchy material (mathematicians call this a Finsler metric)?

The big question mathematicians have asked for over a century is: If you keep looking for new, unique rubber band loops on this weird ball, how fast do you find them as you allow the loops to get longer?

• The Old Answer: For a long time, we knew there were infinitely many loops. But we weren't sure how fast they appeared. The best guess was that they appeared as fast as prime numbers (2, 3, 5, 7, 11...). This is a slow, steady growth.
• The New Answer (This Paper): Bernhard Albach proves that for any "reversible" weird ball (one where going forward is the same as going backward), the loops appear much faster. They appear quadratically.

The Analogy:

• Prime Growth: Imagine finding a new type of tree in a forest. You find one every few miles. It's a slow, steady trek.
• Quadratic Growth: Imagine finding a new type of tree, but the forest is expanding so fast that for every mile you walk, the number of new trees you find doubles, then triples, then quadruples. It's an explosion of variety.

Albach proves that on any such ball, the "tree explosion" (the number of loops) happens at this quadratic rate. This is a massive improvement over previous knowledge.

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How Did He Do It? (The Two-Step Strategy)

The paper is complex, but Albach breaks the problem down into two main scenarios, like a detective solving a case with two different suspects.

Case 1: The "Global Map" Scenario

Imagine the ball has a special, simple loop (a geodesic) that acts like a global map or a turnstile. If you throw a ball at this loop, it bounces off in a predictable way.

• The Math Trick: Albach uses a theorem by a mathematician named Franks. Think of Franks' theorem as a rule about spinning plates. If you have a spinning plate (a map) that preserves area (doesn't stretch or squish the surface) and it has a fixed point (a spot that doesn't move), and it has one other point that keeps coming back to the same spot, then the plate must be spinning in a very specific, chaotic way.
• The Result: This chaos forces the system to create a huge number of new loops. Albach improves Franks' rule to show that this chaos creates loops at a quadratic rate, not just a linear one.

Case 2: The "Two Disjoint Loops" Scenario

What if there is no single "global map" loop? Albach proves that in this case, there must be two separate loops that never touch each other (like two parallel train tracks circling the ball).

• The Math Trick: This is where the paper gets really fancy. Albach lifts the problem from the 2D ball into a 3D world (specifically, a 3-sphere, $S^3$).
  • Imagine the two loops on the ball are actually two rings in a 3D space.
  • He builds a model system: A perfectly symmetrical "toy" sphere with a specific shape (like a dumbbell or a peanut). He calculates exactly how many loops exist on this toy.
  • The "Neck Stretching" Analogy: Imagine you have a real, weird ball and this perfect toy ball. Albach uses a technique called "neck stretching." Imagine pulling the neck of a balloon very thin. As you stretch it, the geometry of the real ball gets forced to look more and more like the toy ball.
  • Because he knows exactly how many loops exist on the toy ball (and he knows they grow quadratically), and because the real ball is forced to mimic the toy ball, the real ball must also have that same quadratic growth.

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Why Does This Matter?

1. It's the "Slowest" Possible Speed: Albach suggests that this quadratic growth is the minimum speed. Even on the most boring, simple-looking weird ball, you can't get away with having fewer loops than this. It's the "speed limit" for chaos on a sphere.
2. No Special Conditions Needed: Previous results required the ball to be "generic" (randomly shaped) or have specific properties. Albach's proof works for any reversible Finsler metric. It's a universal law for these shapes.
3. Connection to Physics: The math used here (Contact Homology) is related to how particles move in magnetic fields and how light travels. Understanding how "loops" (orbits) multiply helps physicists understand complex systems in 3D space.

The "Katok" Warning

The paper mentions a famous counter-example by Katok. If the ball is not reversible (meaning going forward is different from going backward, like a one-way street), you can have a ball with only two loops total.

• Analogy: If you have a one-way street, you can't turn around. You might only have two routes. But if the street is two-way (reversible), you can spin around, loop back, and create an infinite, explosive number of routes.

Summary

Bernhard Albach has solved a 100-year-old puzzle about the number of paths on a sphere. He proved that no matter how weird the sphere is (as long as it's reversible), the number of unique paths you can take grows quadratically.

He did this by:

1. Improving a rule about spinning maps (Franks' Theorem).
2. Building a perfect "toy" sphere and stretching the real sphere to match it (Neck Stretching).
3. Using advanced 3D topology to count the loops.

The result is a fundamental truth about the geometry of our universe: Complexity is inevitable. If you have a two-way path on a sphere, the universe will eventually give you an explosion of new paths.

Technical Summary

Here is a detailed technical summary of the paper "Quadratic Growth of Geodesics on the Two-Sphere" by Bernhard Albach.

1. Problem Statement

The paper addresses the fundamental problem in global Riemannian and Finsler geometry regarding the growth rate of closed geodesics on the 2-sphere ($S^2$).

• Context: For a closed Riemannian surface $M$, let $P_t(g)$ denote the number of geometrically distinct closed geodesics with length at most $t$.
  • If the genus of $M$ is $>1$, the growth is exponential.
  • If the genus is 1 (torus), the growth is at least quadratic.
  • If $M = S^2$, the problem is significantly harder. Historically, Lyusternik and Schnirelmann proved the existence of at least three simple closed geodesics. Bangert and Franks later proved the existence of infinitely many.
• The Gap: Prior to this work, the best known lower bound for the growth rate on $S^2$ (established by Hingston) was that $P_t(g)$ grows at least as fast as the prime numbers:
  $$\liminf_{t\to\infty} \frac{P_t(g)}{\log(t)/t} > 0$$
  This implies a growth rate slightly faster than linear but slower than polynomial.
• Objective: The author aims to prove that for any reversible Finsler metric on $S^2$, the growth rate is actually quadratic:
  $$\liminf_{t\to\infty} \frac{\log(P_t(g))}{\log(t)} \geq 2$$
  This result is sharp, as it is expected to be the slowest possible growth for all such metrics, and it requires no genericity assumptions on the metric.

2. Methodology

The proof is divided into two main cases based on the dynamical properties of the geodesic flow, utilizing tools from symplectic topology, contact geometry, and dynamical systems.

Case 1: Existence of a Global Surface of Section

If the metric admits a closed simple geodesic whose Birkhoff annulus is a global surface of section, the problem reduces to the dynamics of an area-preserving homeomorphism $f$ on an annulus.

• Tool: The author improves upon Franks' Theorem regarding periodic points of area-preserving annulus maps.
• Technique:
  1. Use the existence of a fixed point (from the geodesic) to construct a transverse foliation (using Le Calvez's theory).
  2. Show that the dynamics satisfy a "twisting condition" (rotation numbers of two points straddle a rational number).
  3. Apply Franks' theorem to guarantee the existence of periodic points with specific rotation numbers.
  4. Combine this with a number-theoretic lemma (Lemma 1.4) showing that the number of coprime pairs $(p,q)$ with $q \leq t$ in a fixed interval grows quadratically ($\sim t^2$).

Case 2: Existence of Two Disjoint Simple Closed Geodesics

If no global surface of section exists, Bangert's work implies the existence of two disjoint simple closed geodesics.

• Lifting to $S^3$: The unit tangent bundle of $S^2$ is diffeomorphic to $\mathbb{R}P^3$, which is a quotient of $S^3$. The author lifts the metric to a contact form $\lambda$ on $S^3$. The two disjoint geodesics lift to a link $L$ of Reeb orbits.
• Cylindrical Contact Homology: The core of the proof relies on computing the cylindrical contact homology of the complement of this link in $S^3$.
  • Model System: The author constructs a specific "model" sphere of revolution ($S_m$) and its associated contact form $\lambda_m$. This system is chosen because its geodesics can be explicitly calculated using the Clairaut integral.
  • Morse-Bott Perturbation: Since the model system has degenerate orbits (families of periodic orbits), the author performs a Morse-Bott perturbation to create a non-degenerate contact form $\lambda_S$ while preserving the homology structure.
  • Neck Stretching: Using a "neck stretching" argument (a technique from Symplectic Field Theory), the author proves that if the model system has periodic orbits in specific homotopy classes with bounded action, then the original arbitrary metric $\lambda$ must also possess periodic orbits in those same classes with comparable action bounds.
  • Satellite Orbits: The author classifies closed geodesics on the model sphere as "satellites" of the equator. By analyzing the linking numbers of these orbits with the boundary link $L_1$, they identify specific homotopy classes $y_{(p,q)}$ in the complement of the link.

3. Key Contributions and Results

Theorem 1.1 (Main Result)

For any reversible Finsler metric $g$ on $S^2$, the number of prime closed geodesics $P_t(g)$ satisfies:
$$\liminf_{t\to\infty} \frac{\log(P_t(g))}{\log(t)} \geq 2$$
This establishes a quadratic lower bound, significantly improving the previous logarithmic/prime-number bound.

Theorem 1.2 (Improved Franks' Theorem)

For an area-preserving homeomorphism $f$ of the open annulus isotopic to the identity, if $f$ has at least one periodic point, the number of geometrically distinct periodic orbits with period $\leq t$ grows at least quadratically. This is a crucial dynamical lemma used in the first case of the proof.

Theorem 1.5 (Existence of Orbits in Homotopy Classes)

For a tight contact form on $S^3$ realizing a link transversely isotopic to the model link $L_1$, and for any interval $(a,b)$, there exists a constant $c$ such that for every coprime pair $(p,q)$ with $p/q \in (a,b)$, there exists a closed Reeb orbit in the homotopy class $y_{(p,q)}$ with action bounded by $c \cdot q$.

• This theorem bridges the gap between the explicit model system and arbitrary metrics via the neck stretching argument.

4. Significance and Implications

1. Resolution of a Long-Standing Conjecture: The result confirms that the growth rate of closed geodesics on $S^2$ is at least quadratic, which is the expected "generic" lower bound. It closes the gap between the known existence of infinitely many geodesics and the quantitative rate of their appearance.
2. No Genericity Assumptions: Unlike many results in this field that require the metric to be "bumpy" (non-degenerate) or generic, this proof holds for all reversible Finsler metrics.
3. Connection to Contact Topology: The paper demonstrates the power of Cylindrical Contact Homology and Symplectic Field Theory (SFT) in solving classical problems in Riemannian geometry. The use of "neck stretching" to transfer existence results from a model system to an arbitrary one is a sophisticated application of modern symplectic techniques.
4. Reversibility Constraint: The author notes that the reversibility of the metric is essential. Katok constructed non-reversible Finsler metrics on $S^2$ with only two closed geodesics. This result aligns with Conjecture 1.7 (Hryniewicz), which posits that for contact 3-manifolds, the Reeb flow has either exactly two periodic orbits or a quadratic growth rate.
5. Methodological Advancement: The paper provides a robust framework for analyzing geodesic flows by lifting them to contact flows on $S^3$ and utilizing the topology of the complement of links. The explicit calculation of the contact homology for the model system serves as a template for future studies of similar dynamical systems.

In summary, Albach's paper represents a major breakthrough in the quantitative theory of closed geodesics, proving that the "slowest" possible growth for reversible metrics on the 2-sphere is quadratic, achieved through a deep synthesis of dynamical systems theory and high-dimensional symplectic topology.