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In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary, we introduce several more general classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess certain properties of Strebel differentials and often appear in applications. We discuss the relation between gradient differentials and special signed measures supported on their set of critical trajectories. We provide a characterization of gradient differentials for which there exists a positive measure in the latter class.

Let Z(s) be the Selberg zeta-function associated to a compact Riemann surface. We consider decompositions Z(s) = f(h(s)), where f and h are meromorphic functions, and show that such decompositions can only be trivial.

Given any compact Riemann surface C, there is a canonical meromorphic 2–form η ˆ on C × C , with pole of order two on the diagonal Δ ⊂ C × C , constructed in [4] . This meromorphic 2–form η ˆ produces a canonical projective structure on C. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface C. We prove that these two projective structures differ in general. This is done by comparing the ( 0 , 1 ) –component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The ( 0 , 1 ) –component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in [16] as the Weil–Petersson Kahler form ω w p on the moduli space of curves. We prove that the ( 0 , 1 ) –component of the differential of the section of the moduli space of projective structures corresponding to η ˆ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.

We construct an irreducible holomorphic connection with SL ( 2 , R ) –monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in [5] .

We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi–Yau moduli.

Assume that is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples ( of permutations in the symmetric group , in which and s generate a transitive subgroup G of This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions., 5 pages

Let $X$ be a compact Riemann surface of genus $g \geq 3$. Let $\cat{M}_{Hod}$ denote the moduli space of stable $\lambda$-connections over $X $ and $\cat{M}'_{Hod} \subset \cat{M}_{Hod}$ denote the subvariety whose underlying vector bundle is stable. Fix a line bundle $L$ of degree zero. Let $\cat{M}_{Hod}(L)$ denote the moduli space of stable $\lambda$-connections with fixed determinant $L$ and $\cat{M}'_{Hod}(L) \subset \cat{M}_{Hod}(L)$ be the subvariety whose underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{Hod}$ and $\cat{M}'_{Hod} (L)$, and study their Picard groups. Let $\M_{Hod}(L)$ denote the moduli space of polystable $\lambda$-connections. We investigate the nature of algebraic functions on $\cat{M}_{Hod}(L)$ and $\M_{Hod}(L)$. We also study the automorphism group of $\cat{M}'_{Hod}(L)$.

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we study finite simple groups from a permutation action point of view. We classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most two fixed points on each non-regular orbit and at most four fixed points in total.

A compact Riemann surface is called pseudo-real if it admits anti-conformal (orientation-reversing) automorphisms, but no anti-conformal automorphism of order 2, or equivalently, if the surface is reflexible but not definable over the reals. It is known that there exist pseudo-real surfaces of genus g for every integer g ≥ 2 , and the number of automorphisms of any such surface is bounded above by 12 ( g − 1 ) . In this paper, we extend the concepts of symmetric genus, strong symmetric genus and symmetric cross-cap genus of a group by defining and investigating two new parameters, as follows: (1) the pseudo-real genus ψ ( G ) of a finite group G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which might reverse orientation, and (2) the strong pseudo-real genus ψ ⁎ ( G ) of G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which do reverse orientation, when there exists such a surface for G. Our main theorem is that for every integer g ≥ 2 , there exists a finite group G for which ψ ( G ) = ψ ⁎ ( G ) = g , and hence that the range of each of the functions ψ and ψ ⁎ is the set of all integers g ≥ 2 . We also give an example of a group G for which ψ ⁎ ( G ) is defined but ψ ( G ) ψ ⁎ ( G ) .

It is known that every finitely unbranched holomorphic covering π:$$\pi :\tilde{S} \to S$$ of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding $${{\rm{\Phi}}_{\rm{\pi}}}:Teich\left(S \right) \to Teich\left({\widetilde{S}} \right)$$. By the mutual relations between Strebel rays in Teich(S) and their embeddings in $$Teich\left({\widetilde{S}} \right)$$, we show that the 1st-strata space of the augmented Teichmuller space $$\widehat{Teich}\left(S \right)$$ can be embedded in the augmented Teichmuller space $$\widehat{Teich}\left({\widetilde{S}} \right)$$ isometrically. Furthermore, we show that Φπ induces an isometric embedding from the set Teich(S) B (∞) consisting of Busemann points in the horofunction boundary of Teich(S) into $$Teich{\left({\widetilde{S}} \right)_B}\left(\infty \right)$$ with the detour metric.