Jenkins-Strebel Differentials Research Articles

The surface of circumferences for Jenkins-Strebel Differentials

The surface of circumferences for Jenkins-Strebel Differentials

Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces

It was recently shown that the Caratheodory and Teichmuller metrics on the Teichmuller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmuller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmuller disks on which the two metrics agree, and we conjecture that the Caratheodory and Teichmuller metrics agree on a Teichmuller disk if and only if the Teichmuller 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 Teichmuller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Caratheodory and Teichmuller metrics are different to the case of compact surfaces with punctures.

The asymptotic behavior of Jenkins-Strebel rays

In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem by the author in On behavior of pairs of Teichmüller geodesic rays, 2014 . We also consider the infimum of the asymptotic distance up to choice of base points of the rays along the geodesics. We show that the infimum is represented by two quantities. One is the detour metric between the end points of the rays on the Gardiner-Masur boundary of the Teichmüller space, and the other is the Teichmüller distance between the end points of the rays on the augmented Teichmüller space.

On behavior of pairs of Teichmüller geodesic rays

In this paper, we obtain the explicit limit value of the Teichmüller distance between two Teichmüller geodesic rays which are determined by Jenkins-Strebel differentials having a common end point in the augmented Teichmüller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This is similar to a result of Farb and Masur.

Counting generalized Jenkins–Strebel differentials

We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.

Elementary construction of Jenkins-Strebel differentials

Elementary construction of Jenkins-Strebel differentials

Jenkins–Strebel differentials with poles

Given any compact Riemann surface with finitely many punctures, we show that there exists a unique Jenkins–Strebel differential on the Riemann surface with prescribed heights. In addition, the differential has second order poles at the distinguished punctures with prescribed leading coefficients. As a corollary, we obtain the solution of the moduli problem.

An extremality property of Jenkins-Strebel differentials

Given a compact Riemann surface S with finitely many punctures, in this paper we obtain a new extremality property of a Jenkins-Strebel differential φ on S. As a consequence, we obtain the solutions of several kinds of moduli problems on S.

Teichm�ller spaces of string theory

In Section 4 we saw thatP-lines which are related to Jenkis-Strebel differentials (for example, “harmonic”P-lines) describe a world sheet which has to be created and which has to decay. In Bugajska (1990), 1991) we obtained thatP-line satisfying theP-condition is “associated” to reductions of appropriate holomorphicSL(2, ℂ) bundles (over Riemann surfaces determined by this line) to theSU(2) group. This means (Bugajska, 1990, 1991) that we have to deal withSU(2) bundles over Riemann surfaces equipped with a concrete connectionA. If we interpret this connection as a gauge field of weak interaction (which is responsible for a process of decay), then we see that these completely different approaches yield the same physical situation, namely decay and creation. Moreover, holomorphic quadratic differentials which satisfy theP-condition seem to be just Jenkins-Strebel differentials, or at least most of them (it is still an open question).

On Jenkins-Strebel differentials for open Riemann surfaces

SynopsisDual extremum problems associated with an infinite family of admissible loops on a Riemann surface are shown to be solvable by Jenkins-Strebel differentials. Then the inequalities associated with these problems are used to calculate the first derivatives of extremal length functionals on Teichmüller space and to estimate the difference quotients for the second derivatives of these functions.