Composition Operators on Compact Riemann Surfaces

In the present paper, the characterization of invertible composition oper-ators on compact Riemann surfaces is obtained, which is very different from that in the case of Hardy or Bergman spaces on a disk of the complex plane C.

SOME FURTHER RESULTS FOR HOLOMORPHIC MAPS ON PARABOLIC RIEMANN SURFACES

In [1],they generalized R.Nevanlinna’s results to Y,where Y is a parabolic Riemann Surface.In this paper,following their method,we develop some further results for holomorphic maps on Y,including the maps into Pn(C),the complex projective varieties,and Abelian varieties.

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification

Let M be an open Riemann surface and G:M→Pn(C)be a holomorphic map.Consider the conformal metric on M which is given by ds^(2)=‖■‖^(2m)|w|^(2),where■is a reduced representation of G,ωis a holomorphic 1-form on M and m is a positive integer.Assume that ds^(2) is complete and G is k-nondegenerate(0≤k≤n).If there are q hyperplanes H1,H2,…,Hq■Pn(C)located in general position such that G is ramified over Hj with multiplicity at leastγj(>k)for each j∈{1,2,…,q},and it holds that■,then M is flat,or equivalently,G is a constant map.Moreover,one further give a curvature estimate on M without assuming the completeness of the surface.

Fredholm Composition Operators on Riemann Surfaces

It is proved that the invertibility of a composition operator on the differential form space for a Riemann surface is equivalent to its Fredholmness. In addition, the Fredholmness of weighted composition operators is discussed.

A Trudinger-Moser Inequality Involving Lp-norm on a Closed Riemann Surface

In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.

Twists and Gromov Hyperbolicity of Riemann Surfaces

The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces （with their Poincare metrics）. We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.

Bers-Orlicz Spaces on the Product Riemann Surface

In this paper,the Bers-Orlicz spaces on the automorphic form A_α■(G)(or EA_α■(G)) and L_α■(G)(E_α■(G))on the product Riemann surfaces are studied.We prove that each f ∈A_α■(G) is a cusp form.For f ∈A_α■(G),we give the reproducing formula.And,we give the projective operator P_αfrom L_α■(G) to A_α■(G)(or E_α■(G) to EA_α■(G)).After giving some fundamental properties of the Poincaréseries,we prove a dual theorem A_α■(G)=(EA_α■(G))~*.

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W^(1,2)(Σ,g),∫_(θΣ)uds_(g)=0,u≠0∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)/∫_(θΣ)u^(2)ds_(g)and H={u∈W^(1,2)(Σ,g):∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)-α∫_(θΣ)u^(2)ds_(g)≤1 and∫_(θΣ)uds_(g)=0},where W^(1,2)(Σ,g)denotes the usual Sobolev space and▽g stands for the gradient operator.By the method of blow-up analysis,we obtain sup_(u∈H)∫_(θΣ)e^(πu^(2))ds_(g){<+∞,0≤α﹤λ_(1)(∂Σ),=+∞,α≥λ_(1)(∂Σ)Moreover,the author proves the above supremum is attained by a function u∈H∩C^(∞)(∑)for any 0≤α<λ_(1)(θΣ).Further,he extends the result to the case of higher order eigenvalues.The results generalize those of[Li,Y.and Liu,P.,Moser-Trudinger inequality on the boundary of compact Riemannian surface,Math.Z.,250,2005,363–386],[Yang,Y.,Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J.Math.,227,2006,177–200]and[Yang,Y.,Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two,J.Diff.Eq.,258,2015,3161–3193].

A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group

A Riemann surface S having field of moduli M,but not a field of definition,is called pseudo-real.This means that S has anticonformal automorphisms,but none of them is an involution.A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d≥4 in P^2/C.We characterize pseudo-real-plane Riemann surfaces»S,whose conformal automorphism group Aut+(S)is PGL3(C)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of P^2/C.In particular,we show that such pseudo-real-plane Riemann surfaces exist only if Aut+(S)is cyclic of even order n dividing the degree d.Explicit families of pseudo-reai-plane Riemann surfaces are given for any degree d=2pm with m>1 odd,p prime and n=d/p.

Finite Degree Holomorphic Covers of Compact Riemann Surfaces

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.

The Asymptotic Behavior of Chern-Simons Higgs Model on a Compact Riemann Surface with Boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter A 〉 0 has at least two solutions （uλ^-, uλ^2） for A sufficiently large, which satisfy that uλ^1 - -u0 almost everywhere as λ →∞, and that uλ^2 →-∞ almost everywhere as λ→∞, where u0 is a （negative） Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ →∞, and prove that uλ^2 - uλ^2- converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary OM is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.

Solutions for Toda System on Riemann Surface with Boundary

In this paper, we study the solutions for Toda system on Riemann surface with boundary. We prove a sufficient condition for the existence of solution of Toda system in the critical case.

Gromov Hyperbolicity of Riemann Surfaces

We study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its ＂building block components＂. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.

Automorphism Groups of Pseudo-real Riemann Surfaces of Low Genus

A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution.We obtain the classifcation of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2,3 and 4.For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is eitherC4orC8or the Fro¨benius group of order 20,and in the case ofC4there are exactly two possible topological actions.Let MK P R,g be the set of surfaces in the moduli space MK g corresponding to pseudo-real Riemann surfaces.We obtain the equisymmetric stratifcation of MK P R,g for generag=2,3,4,and as a consequence we have that MK P R,gis connected forg=2,3 but MK P R,4has three connected components.

紧黎曼曲面上锥度量的高斯博内公式

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.

On the linearization of the coupled Harry-Dym soliton hierarchy

This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.

THE DISCUSSION ON THE NORMAL FUNDAMENTAL REGION OF FUCHS GROUP WITH NONEUCLIDEAN GEOMETRY

We define the fundamental region homeomorphic to the corresponding Riemann surface according to the methods on form-conserved circle of the fractional linear transformation in explaining that the form-conserved circle is the perpendicular bisector of noneuclidean segment limited by the end points both the origin and the equivalent point by the same transformation just mentioned and, consequently, its sense on noneuclidean geometry is clarified.The result does not appear in current literatures and is useful for the research of superstring.

SOLVING δ_b ON PARABOLIC LAMINATIONS

Let X be a compact set which is laminated by parabolic Riemiann surfaces. For the CR positive line bundle L, there exists an integer N ∈ N such that for any s 〉 N and any continuous v ∈∧^（0,1）X×L^×s, there exists a continuous u ∈ L^×s solving δb^-u = v.