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
Let M be a compact Riemann surface, h(x) a positive smooth function on M, and Ф(x) a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ(u)=1/2∫M|△u|2eФdVg+...Let M be a compact Riemann surface, h(x) a positive smooth function on M, and Ф(x) a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ(u)=1/2∫M|△u|2eФdVg+8πc∫MueФ-8πclog∫Mheu+ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ(X).展开更多
Let M be an open Riemann surface with a finite set of punctures, a complete Poincar(?)-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle...Let M be an open Riemann surface with a finite set of punctures, a complete Poincar(?)-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle, and the existence of a Hermitian-Einstein metric on the bundle is established.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
It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively....It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S/{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston's construction.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
Let S be a Riemann surface that contains one puncture x. Let be the collection of simple closed geodesics on S, and let denote the set of mapping classes on S isotopic to the identity on S U {x}. Denote by tc the ...Let S be a Riemann surface that contains one puncture x. Let be the collection of simple closed geodesics on S, and let denote the set of mapping classes on S isotopic to the identity on S U {x}. Denote by tc the positive Dehn twist about a curve c ∈ . In this paper, the author studies the products of forms (tb^-m o t^na) o f^k, where a, b ∈ and f ∈ . It is easy to see that if a = b or a, b are boundary components of an x-punctured cylinder on S, then one may find an element f ∈ such that the sequence (tb^-m o t^na) ofk contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the sequence (tb^-m o t^na) o f^k contains infinitely many powers of Dehn twists, then a, b must be the boundary components of an x-punctured cylinder on S and f is a power of the spin map tb^-1 o ta.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
In this paper,we give some conditions on the surjective of multiply maps H^0(R,L)×H^0(R,K)→H^0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.
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.
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.
文摘Let M be a compact Riemann surface, h(x) a positive smooth function on M, and Ф(x) a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ(u)=1/2∫M|△u|2eФdVg+8πc∫MueФ-8πclog∫Mheu+ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ(X).
基金Project surpported partially by the National Natural Science Foundation of China (Grant No. 19701034).
文摘Let M be an open Riemann surface with a finite set of punctures, a complete Poincar(?)-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle, and the existence of a Hermitian-Einstein metric on the bundle is established.
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.
文摘It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S/{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston's construction.
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))~*.
文摘Let S be a Riemann surface that contains one puncture x. Let be the collection of simple closed geodesics on S, and let denote the set of mapping classes on S isotopic to the identity on S U {x}. Denote by tc the positive Dehn twist about a curve c ∈ . In this paper, the author studies the products of forms (tb^-m o t^na) o f^k, where a, b ∈ and f ∈ . It is easy to see that if a = b or a, b are boundary components of an x-punctured cylinder on S, then one may find an element f ∈ such that the sequence (tb^-m o t^na) ofk contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the sequence (tb^-m o t^na) o f^k contains infinitely many powers of Dehn twists, then a, b must be the boundary components of an x-punctured cylinder on S and f is a power of the spin map tb^-1 o ta.
基金the National Natural Science Foundation of China (Grant No. 10671093)the Scientific Research Starting Foundation for Returned Overseas Chinese Scholars, Ministry of Education,China, and NSA (Grant No. MSPR-06G-026)
文摘In this paper, we prove a uniqueness theorem for algebraic curves from a compact Riemann surface into complex projective spaces.
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.
文摘In this paper,we give some conditions on the surjective of multiply maps H^0(R,L)×H^0(R,K)→H^0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.
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.