Meromorphic quadratic differentials with complex residues and spiralling foliations

Abstract

A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that for a fixed Riemann surface, any such foliation is realized by a quadratic differential with second order poles at marked points. Furthermore, such a differential is uniquely determined if one prescribes complex residues at the poles that are compatible with the transverse measures around them. This generalizes a theorem of Hubbard and Masur concerning holomorphic quadratic differentials on closed surfaces, as well as a theorem of Strebel for the case when the foliation has only closed leaves. The proof involves taking a compact exhaustion of the surface, and considering a sequence of equivariant harmonic maps to real trees that do not have a uniform bound on total energy.

Cutting orientable ribbon graphs

“…Definition : For M a surface with boundaries measurable foliations on M can be define in several possible ways. Some references on the subject are for instance [1],[fathi2012thurston], [8]. Let MF(M ) as the space of foliations on M • such that the one form that locally define the foliation is given (locally) by the real part of a quadratic differential with a simple poles at the puncture.…”

Holomorphic quadratic differentials in Teichmüller theory

“…The induced singular-flat metric is what we call a half-plane structure comprising Euclidean half-planes attached to the critical graph. The choice of the principal part P determines the singular-flat geometry of the resulting "planar end" around the puncture (see [52]).…”

An infrared bootstrap of the Schur index with surface defects

“…However, here we are varying the UV moduli such as masses and couplings as well, and therefore, these singularities should be understood as loci in the enlarged moduli space. 54 In fact, the existence of a sequence of differentials that implements a Whitehead move can also be argued through the theory of half-translation surfaces, for background see [119,120]. One way of specifying a holomorphic quadratic differential on a surface is to describe charts to C that differ by half-translations.…”