An exact mathematical rule links the capacity of an ellipsoid’s unit disk bundle to two special geodesic lengths. The preprint says that for every ellipsoid E(a,b,c) with ordered positive semiaxes, a≥b≥c>0, the Hofer–Zehnder capacity is the smaller of twice the systole and the length of a shortest closed geodesic with Morse index 3. The claim is an equality for the whole stated class.
That makes the result a kind of geodesic selector. A closed geodesic is a path that returns to its starting point; in this setting, the systole is the distinguished length realized by the minor principal ellipse. The competing path is the shortest geodesic with Morse index 3, the label used in the analysis for the selected class. The capacity follows whichever of those two quantities is smaller.
The choice changes with the ellipsoid’s shape. The selected index-3 geodesic can be the major principal ellipse or, in the oblate case, a simple geodesic winding once around the z-axis and intersecting the major principal ellipse four times. The paper plots the parameter regions where twice the systole is shorter and where the index-3 length is shorter, after scaling a=1.
How the equality is built
To obtain the lower side of the result, the authors use Riemannian billiards. Their general construction approximates billiard dynamics by smooth Hamiltonians and expresses the lower bound through the shortest admissible periodic billiard trajectory. For the ellipsoid class, this gives a capacity at least as large as the smaller of the shortest index-3 geodesic length and twice the systole.
This lower bound already has the shape of the final answer. It says the capacity cannot fall below the smaller of the two candidate lengths, but it does not alone establish equality. The second half comes from upper bounds.
That billiard route also carries a caveat. The preprint says its notion of admissibility may depend on the chosen approximation scheme, and it leaves open whether admissibility is independent of those choices. The question is attached to the construction used for the lower bound.
One upper-bound argument uses pseudoholomorphic curves and neck-stretching. In the ellipsoid setting, it proves that the capacity is no greater than the length of the shortest Morse-index-3 geodesic. This isolates the same geodesic that appears in the minimum.
A second upper-bound construction uses the pair-of-pants product in symplectic homology and the filtered Viterbo isomorphism, which corresponds to the Chas–Sullivan loop product. The two methods supply complementary controls on the geodesic quantities entering the ellipsoid formula.
For any Riemannian metric on a two-sphere, the symplectic-homology bound is that the capacity of the associated unit disk bundle is no more than twice the diastole, another geodesic-length quantity. Under positive curvature, the diastole equals the systole, so the bound becomes twice the systole.
Beyond the ellipsoid
The preprint also states an upper bound for positively curved Riemannian two-spheres. It uses qualifying closed geodesics of Morse index 3 and separates them by their self-intersection pattern. In even-self-intersection cases, the geodesic length is at most 2πR. In odd-self-intersection cases, its length plus the systole is at most 2πR, where R is the circumradius used in the bound.
These are bounds under stated geometric conditions, not a second exact capacity formula for every two-sphere. The exact equality in the paper is tied to the ellipsoid unit disk bundles, while the broader two-sphere statements limit the capacity from above.
The boundary is explicit. The introduction says an extension to all closed surfaces of constant curvature remains unresolved. It notes that capacity is unknown in general for flat Klein bottles, and that even finiteness is open for closed orientable hyperbolic surfaces.
The numerical comparison should therefore be read as a map of the covered ellipsoid parameter space, not as a result about arbitrary surfaces. It shows where twice the systole or the index-3 length is shorter after the normalization a=1.
A theorem with a defined boundary
This is a theoretical theorem about specified symplectic and Riemannian objects, not an empirical estimate. The exact statement has no statistical uncertainty; its scope is set by the ordered ellipsoid class, while the two-sphere results are bounds under their stated conditions.
The paper is identified as arXiv version 1, dated 20 Aug 2026, and as a preprint.
Within those limits, the result offers a precise selector: for every ellipsoid in the stated class, the capacity is determined by the shorter of twice the systole and a shortest Morse-index-3 geodesic, with billiards supplying the lower bound and the two complementary theories supplying upper bounds.
Paper data and sources
Original title: Hofer-Zehnder capacity as a geodesic selector
Authors: Johanna Bimmermann, Beomjun Sohn
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text