Isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$

TL;DR

This work analyzes the isoperiodic foliation of the genus-1 stratum ΩM1(1,1,-2), proving that every nontrivial period character is realized by a unique leaf and that leaves are translation surfaces with a rich wall–chamber structure. It classifies the Veech groups arising from leaves, identifies a Loch Ness monster topology for the metric completions of marked leaves, and delineates positive, negative, and real-arithmetic leaf types, including how these types degenerate toward ΩM1(2,-2) and the boundary. The authors develop a detailed conformal-geometry framework, connecting leaves to Teichmüller space, and provide explicit chamber-gluing constructions, degeneration rules, and GL^+(2,R) actions to describe large-scale geometry and dynamics. The results give a comprehensive picture of the isoperiodic foliation in this meromorphic setting, with precise descriptions of chamber adjacencies, degenerations, and Veech-group constraints, contributing to the broader understanding of translation surfaces with poles. The findings have implications for moduli of meromorphic differentials, explicit Teichmüller-theoretic maps, and the geometric structure of leaves in low-genus strata, informing both foliation theory and the study of flat surfaces with singularities.

Abstract

This paper describes the geometry and topology of leaves of the isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$. We prove that each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface and supports a singular Euclidean structure whose singularities correspond to isoperiodic forms in the lower stratum $Ω\mathcal{M}_1(2,-2)$. Along the way we also characterize the possible groups arising as the Veech group of a leaf and give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.

Figures (41)

• Figure 1: Metric completion of an isoperiodic leaf $\mathfrak{ML}(\,\rho,\mu\,)$.
• Figure 2: A schematic picture of an unmarked positive leaf, drawn in the upper half plane. For the chamber based at $\infty$, the two ends of its boundary goes to infinity with asymptotics $\asymp\log x$.
• Figure 3: Translation surface in the marked stratum $\mathcal{H}_1(1,1,-2)$ with degenerate core. Such a structure can be realized by gluing sides with same color.
• Figure 4: Translation surface in the marked stratum $\mathcal{H}_1(1,1,-2)$ whose core is a slit torus. On the left it is depicted the core and, on the right, it is depicted the domain of the pole. Note this latter is a slit copy of $\mathbb C$ whose point at infinity is the double pole of the resulting structure. In this case $b=2$ because, in the slit plane, there are two saddle connections.
• Figure 5: Translation surface in the marked stratum $\mathcal{H}_1(1,1,-2)$ whose core is a cylinder. Such a structure can be realized by gluing sides with same color. On the left it is depicted the core and, on the right, it is depicted the domain of the pole. As above, this latter is a slit copy of $\mathbb C$ whose point at infinity is the double pole of the resulting structure.

Theorems & Definitions (148)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4: Local geometry around a singularity