arXiv subjects

Ser-Wei Fu

Publications and source records attributed to Ser-Wei Fu.

4 recordsLinked to original sources

arXiv2020

Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q < 3 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case q > 2 and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if q > 2 and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

math.GT↗

arXiv2019

Cusp excursions for the earthquake flow on the once-punctured torus

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow and the horocyclic flow behave very differently in cusp excursions. In particular, we prove a relation between the systole function and continued fractions and discuss the cusp excursions of earthquake trajectories.

math.GT↗

arXiv2018

Flat grafting deformations of quadratic differentials on surfaces

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. We use flat grafting to construct paths connecting any pair of quadratic differentials. In other words, we characterize cone point splitting deformations. The slices of quadratic differentials closed under flat grafting maps with a fixed direction arise naturally and we prove rigidity properties with respect to the lengths of closed curves.

math.GT↗

arXiv2013

Length spectra and strata of flat metrics

In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Leininger-Rafi. Specifically, for any stratum with more zeroes than the genus, the length spectrum of a set of simple closed curves determines the flat metric in the stratum if and only if the set is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the length spectrum of a finite set of closed curves.

math.GT↗