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 comple...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.展开更多
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),...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.展开更多
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...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.展开更多
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...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.展开更多
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 ...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.展开更多
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 Trud...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.展开更多
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 i...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.展开更多
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 su...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.展开更多
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■(...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))~*.展开更多
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...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].展开更多
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 plan...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.展开更多
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 ...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.展开更多
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 lea...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.展开更多
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.
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 hyper...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.展开更多
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.F...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.展开更多
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 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.展开更多
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 f...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.展开更多
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...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.展开更多
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, t...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.展开更多
文摘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.
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.12101068,12261106,12171050)。
文摘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.
基金Supported by National Natural Science Foundation of China
文摘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.
基金Supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China。
文摘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.
基金Supported partially by National Natural Science Foundation of China
文摘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.
文摘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.
基金Supported by the National Nature Science Foundation of China.
文摘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))~*.
基金supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China
文摘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].
文摘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.
基金the Center of Excellence"Geometric Analysis and Mathematical Physics"of the Academy of Finland
文摘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.
基金Supported by National Natural Science Foundation of China (Grant Nos.10701064,10931001)XINXING Project of Zhejiang University
文摘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.
基金Supported by National Natural Science Foundation of China (Grant No. 11001268)
文摘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.
基金supported by a grant from DGI(BFM 2003-04870)Spainsupported by a grant from DGI(BFM 2000-0022)Spain
文摘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.
基金Supported by Spanish Government Research(Grant No.MTM2011-23092)
文摘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.
基金Support by the Project of Stable Support for Youth Team in Basic Research Field,CAS(Grant No.YSBR-001)NSFC(Grant Nos.12271495,11971450 and 12071449).
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant No 10471132), and the National Key Basic Research Special Foundation of China (Grant No 113000531034).Acknowledgments The authors are obliged to the anonymous referee for his valuable remarks and suggestions.
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(11271359)
文摘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.