An arXiv preprint reports that several specified classes of translation surfaces can be continuously reshaped until they reach a maximal surface, while their systole—the length of a shortest saddle connection in this setting—rises strictly along the route. The result covers square-tiled surfaces and surfaces obtained from regular hexagons or regular octagons, within a fixed stratum and with unit-area normalization used for the comparison.
In the paper’s constructions, the surfaces are built from polygonal pieces whose boundary sides retain their prescribed identifications as the shapes change. The central question is whether a non-global-maximal surface can be connected to a maximal one by a continuous path on which the systole is strictly larger at every later parameter.
The answer is affirmative for the classes the paper studies. It is not a claim that every non-maximal translation surface admits such a route.
A result about routes, not all surfaces
The paper organizes its objects into three named classes: P for square-tiled surfaces, Q for surfaces obtained from regular hexagons, and R for surfaces obtained from regular octagons. Its main theorem states that each surface in P, Q or R has a continuous path in its stratum, indexed from 0 to 1, that ends at a maximal surface with strict systole ordering.
That is an existence result for specified mathematical families. The supplied analysis describes no empirical sample: the objects are defined translation surfaces and deformation families, and the evidence comes from explicit geometric constructions and their proofs.
The paper also identifies an obstruction. A local maximum of the systole that is not the global maximum cannot be the starting point of a within-stratum deformation along which the systole increases monotonically. The supplied text does not characterize every possible obstruction.
How the deformations work
The construction changes the component polygons continuously while preserving their side lengths, the combinatorial arrangement of their boundaries and the original side identifications. The family remains in the same stratum. The surfaces are then normalized to unit area, so systole lengths can be compared at a common scale.
The monotonicity claims come from explicit calculations rather than statistical estimates. The analysis describes trigonometric area calculations, determinants and the shoelace formula, together with arguments based on the sign of derivatives. The Mean Value Theorem is used to establish monotonicity on closed intervals.
For the square-tiled case, the paper uses a rhombus deformation. The path is continuous and strictly systole-increasing; at its terminal configuration, the rhombi split into equilateral triangles, and the endpoint is maximal.
Hexagons and octagons meet the square-tiled case
The regular-hexagon construction follows a continuous family of equilateral hexagons with fixed unit side length. The family runs from a regular hexagon to the boundary of two adjacent unit squares. Along the deformation, the paper reports decreasing polygonal area and, after unit-area normalization, a strict increase in the systole.
The hexagon path ends at a square-tiled surface. It is then concatenated with the square-tiled path to a maximal surface, while retaining strict systole increase along the combined route.
The regular-octagon construction uses a similar type of family: equilateral octagons with fixed unit side length are continuously deformed until they reach the boundary of three adjacent unit squares. The paper reports strict systole ordering along this deformation parameter.
This path also ends at a square-tiled surface and is joined to the square-tiled route to a maximal surface. The combined octagon-to-maximal path therefore retains strict systole increase throughout the construction described in the paper.
Where the theorem stops
The paper extends the construction to surfaces that can be cut along shortest saddle connections into pieces made only from squares, regular hexagons or regular octagons. In that extension, the polygons have side length equal to the original systole. The polygonal-decomposition condition is part of the result; it is not presented as a statement about arbitrary surfaces.
The restriction is central. The study does not establish that every non-maximal translation surface can be moved to a maximal one with a steadily increasing systole. Nor does it show that a non-global local maximum can be escaped by such a monotone path.
Because the work is theoretical, it provides no empirical sample, statistical uncertainty or independent numerical validation. Its conclusions depend on remaining within a fixed stratum and using unit-area normalization, as well as on the geometric constructions described in the paper.
The supplied analysis also notes that the characterization of maximal surfaces cited in the paper comes from an earlier reference and was not independently assessed in this review. Broader evaluation would require examining that reference and the wider translation-surface literature.
An open problem beyond the polygons
The document is an arXiv version 1 preprint dated 20 August 2026. Its supplied metadata lists an institutional affiliation but no funding source.
The next questions are how far the method can travel beyond these constructions: which additional classes admit strictly systole-increasing paths, whether arbitrary non-maximal surfaces outside the stated polygonal families can be handled, and how the obstruction behaves across all strata.
Paper data and sources
Original title: Systole Increasing Deformations to Maximal Translation Surfaces
Authors: Achintya Dey, Bidyut Sanki
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text