Rigidity in the Planar Ulam Floating Body Problem with perimetral density $σ=\tfrac16$

Abstract: We study the two-dimensional Ulam's floating body problem for convex domains with perimetral density $σ=\tfrac16$. Using the framework of Zindler carousels, we reduce the problem to a two-dimensional dynamical system associated with an inscribed equilateral hexagon. Our main result shows that the disk is the only convex domain floating in equilibrium in every position for this perimetral density. This provides a new rigidity result for rational perimetral densities in the convex setting.

• The paper establishes that the disk is the unique convex solution for the σ = 1/6 floating body problem through a reduction to a Hamiltonian system.
• It employs a novel Zindler carousel framework and central symmetry conditions to constrain the dynamics of inscribed equilateral hexagons.
• The analysis uses precise period estimates and geometric constraints to rule out any non-circular, strictly convex bodies.

Rigidity in the Planar Ulam Floating Body Problem for Perimetral Density $\sigma = \frac{1}{6}$

Introduction and Background

The planar Ulam floating body problem asks whether a solid of uniform density that floats in equilibrium in every position must necessarily be a disk. For a two-dimensional domain $K \subset \mathbb{R}^2$, this posits a geometric restriction on its cross-section when immersed in a fluid: the so-called "perimetral density" $\sigma$ prescribes the relative arc lengths of $K$ submerged above and below any given waterline. For $\sigma = \frac{1}{2}$, non-circular solutions exist (the Zindler curves), while previous rigidity results for rational densities such as $\sigma = \frac{1}{3}$ and $\sigma = \frac{1}{4}$ indicate the disk is unique in the strict convex case.

The paper investigates the next nontrivial rational instance: $\sigma = \frac{1}{6}$. It proves a rigidity theorem for convex planar domains, asserting that the disk is the unique (up to congruence) convex body floating in equilibrium in every orientation for this perimetral density.

Zindler Carousels and the Dynamical Systems Reduction

A key tool is the equivalence between the existence of a strictly convex floating body with rational perimetral density $\sigma = \frac{1}{N}$ and the existence of a Zindler carousel: a parameterized family of inscribed equilateral $N$-gons, whose side length is invariant with respect to the carousel parameter. For convex bodies, each such $K \subset \mathbb{R}^2$0-gon yields an associated system of geometric ODEs derived from the evolution of the polygon's internal angles as it is rolled along the boundary. The $K \subset \mathbb{R}^2$1 case (hexagon) is of special interest for $K \subset \mathbb{R}^2$2.

The key step reduces the problem to the analysis of a two-dimensional Hamiltonian system for the internal angles of the hexagon, followed by geometric constraints that preclude the existence of non-circular solutions.

Central Symmetry and Parametric Reduction

The analysis reveals that, under the floating equilibrium hypothesis for $K \subset \mathbb{R}^2$3, the inscribed equilateral hexagons must be centrally symmetric at all positions. This property, combined with the "carousel midpoint" condition, forces the boundary itself to possess central symmetry about a fixed point (and not just the instantaneous center of the hexagon).

(Figure 1)

Figure 1: Level curve of the Hamiltonian $K \subset \mathbb{R}^2$4 in the reduced phase space.

The central symmetry reduces the dynamical system for the three independent interior angles to a $K \subset \mathbb{R}^2$5 Hamiltonian system with Hamiltonian $K \subset \mathbb{R}^2$6. The total symmetry of a disk arises as a singular orbit at the maximum of $K \subset \mathbb{R}^2$7.

Geometric Dynamical Systems and Hamiltonian Structure

The reduction yields the ODE system: $K \subset \mathbb{R}^2$8 with $K \subset \mathbb{R}^2$9 constant along orbits.

For strict convexity, the internal angles $\sigma$0 are constrained to $\sigma$1, and the symmetry relations further restrict the allowable phase space, inducing concavity of $\sigma$2 and a unique maximum at $\sigma$3, the regular hexagon (corresponding to the disk).

The resulting orbits are closed level curves of $\sigma$4 inside a strictly convex allowed set. Non-disk orbits correspond to nontrivial closed (periodic) oscillations.

Period Estimates and Nonexistence

The period $\sigma$5 of angle oscillations for orbits inside the allowed region is bounded both above and below by explicit estimates derived via careful integral bounds and convexity considerations: $\sigma$6 Combined with perimeter constraints for any convex figure embedding such hexagons, a contradiction is forced for any putative non-disk, because the required period quantization (from rotational symmetry) cannot be fulfilled without violating the period estimates.

Through this analytic machinery, the disk emerges as the unique solution—no nontrivial periodic orbits exist that comply with all geometric and dynamical constraints for $\sigma$7.

Implications and Further Directions

The result closes a gap in the family of rational perimetral densities where previous approaches had stopped at $\sigma$8, confirming that rigidity persists for $\sigma$9 in the convex case. The reduction to a hyper-elliptic Hamiltonian system, the identification of global geometric constraints, and the precise period analysis yield an approach that could, in principle, be extended to higher rational densities, but with rapidly increasing technical complexity.

The analysis also exposes structural differences versus the exceptional $K$0 case, where the integrable system is much more flexible and supports diverse (non-circular) floating bodies.

An open avenue remains the full characterization of monotonicity and period function properties for the reduced Hamiltonian system in general $K$1-gon settings. Further, leveraging the Hamiltonian perspective for higher $K$2 values or potentially generalizing to higher dimensions is an interesting direction for future research.

Conclusion

This work provides a definitive rigidity result for the planar Ulam floating body problem with perimetral density $K$3: the disk is the only strictly convex planar domain (with $K$4 boundary) that floats in equilibrium in all orientations under this density. The approach combines the Zindler carousel reduction, central symmetry arguments, and Hamiltonian dynamical systems methods. Extending these methods to other rational densities or higher dimensions remains an open and technically challenging problem (2604.10330).