Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1711.04722

Classifying complex geodesics for the Carath\'eodory metric on low-dimensional Teichm\"uller spaces

Dmitri Gekhtman·Vladimir Markovic
View original source ↗Read source document ↗Export metadata ↓

Abstract

It was recently shown that the Carath\'eodory and Teichm\"uller metrics on the Teichm\"uller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichm\"uller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichm\"uller disks on which the two metrics agree, and we conjecture that the Carath\'eodory and Teichm\"uller metrics agree on a Teichm\"uller disk if and only if the Teichm\"uller disk is generated by a differential with no odd-order zeros. Using dynamical results of Minsky, Smillie, and Weiss, we show that it suffices to consider disks generated by Jenkins-Strebel differentials. We then prove a complex-analytic criterion characterizing Jenkins-Strebel differentials which generate disks on which the metrics coincide. Finally, we use this criterion to prove the conjecture for the Teichm\"uller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Carath\'eodory and Teichm\"uller metrics are different to the case of compact surfaces with punctures.

Explore related subjects

math.GTmath.CV

Keep this discovery

BibTeXRIS

Dmitri Gekhtman, Vladimir Markovic. 2017-11-13. Classifying complex geodesics for the Carath\'eodory metric on low-dimensional Teichm\"uller spaces. https://arxiv.org/abs/1711.04722

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗
arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗