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Register a new userDrawing cone spherical metrics via Strebel differentials
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Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on compact Riemann surfaces by drawing on the surfaces some class of connected metric ribbon graphs.
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In this manuscript, by using Belyi maps and dessin denfants, we construct some concrete examples of Strebel differentials with four double poles on the Riemann sphere. As an application, we could give some explicit cone spherical metrics on the Riemann sphere.
Strebel differentials are a special class of quadratic differentials with several applications in string theory. In this note we show that finding Strebel differentials with integral lengths is equivalent to finding generalized Argyres-Douglas singularities in the Coulomb moduli space of a U(N) $N=2$ gauge theory with massive flavours. Using this relation, we find an efficient technique to solve the problem of factorizing the Seiberg-Witten curve at the Argyres-Douglas singularity. We also comment upon a relation between more general Seiberg-Witten curves and Belyi maps.
A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 pi mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X geq 2$. As an application, we prove the following two results about irreducible metrics: $bullet$ as $g_X geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $geq 2(d+3-3g_X)$ in ${rm Sym}^d(X)$; $bullet$ as $g_X geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$.
A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential.
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