A note on harmonic quasiconformal mappings

In this note we show that a harmonic quasiconformal mapping f=u+iv with respect to the Poincaré metric of the upper half plane onto itself such that v(x,y)=v(y) or u(x,y)=u(x) is a conformal mapping.     

The ACL property of homeomorphisms under weak conditions

Suppose thatD is a domain in andy=f(x)∶D→R ⁿ is a homeomorphism. We prove that if the modulus dilatationK(x, f) satisfies the condition A thenf(x) is ACL. Project supported by the National Natural Science Foundation of China and JTU     

The fixed subgroups of homeomorphisms of Seifert manifolds

Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold B M,and fπ : π1(M) →π1(M) be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of fπ, namely, rank Fix(fπ) 2rankπ1(M),which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.     

Topological Loewner theory on Riemann surfaces

We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.     

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 Poincaré 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.     

Beurling-Ahlfors延拓与调和映照

研究BA延拓和调和映照的关系,首先,给出了BA延拓为双曲调和的一个必要条件,特别地,若边界对应h局部是C~2和奇的,则其BA延拓不是双曲调和的。其次,证明了若h是分段C~2的则其BA延拓不是π调和的,除非h(x)=ax b,x∈R.     

Triangulations and homology of Riemann surfaces

We derive an algorithmic way to pass from a triangulation to a homology basis of a (Riemann) surface. The procedure will work for any surfaces with finite triangulations. We will apply this construction to Riemann surfaces to show that every compact hyperbolic Riemann surface has a homology basis consisting of curves whose lengths are bounded linearly by the genus of and by the homological systole.

This work got started by comments presented by Y. Imayoshi in his lecture at the 37th Taniguchi Symposium which took place in Katinkulta near Kajaani, Finland, in 1995.

     

紧致黎曼面上具有可积伸张及指定界值的拟共形同胚的极值映照

设S是亏格大于1的紧致黎曼面，本文对S上的拟共形ID—同胚定义了一种极值，证明了在每一个S到另外一紧致字文面S’的保向同胚同伦类中，如果存在一个广义的Teichmuller映照，那么在这种新定义的极值意义下，它是唯一最小的极值映照．     

Composition operators on Hardy spaces of Riemann surfaces

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.     

On multiply of sections of line bundles on compact Riemann surfaces

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 note on uniformization of Riemann surfaces by Ricci flow

We clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.     

On locally biholomorphic finitely valent mappings from multi-connected domains onto the open disc

In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the valentness m of locally biholomorphic mappings from multi-connected domains onto the open unit disc.     

Distortion theorems of plane quasiconformal mappings

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φK(r) and λ(K). By using Teichmüller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.     

Uniqueness theorem for algebraic curves on compact Riemann surfaces

In this paper, we prove a uniqueness theorem for algebraic curves from a compact Riemann surface into complex projective spaces.     

Divisor spaces on punctured Riemann surfaces

In this paper, we study the topology of spaces of -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

     

The invariance of inner and outer linearly locally connected sets under quasiconformal mappings

In this paper, we prove that the inner and outer linearly locally connected sets of Rn are invariant respectively under quasiconformal mappings which fixed the point at infinity. On the other hand, we prove the converses of above results to be true also.     

A Schwarz-Pick inequality for harmonic quasiconformal mappings and its applications

The main result of this paper is the sharp generalized Schwarz-Pick inequality for euclidean harmonic quasiconformal mappings with convex ranges, which generalizes a result given by Mateljevi?. As its applications, we obtain the property of quasi-isometry with respect to the Poincaré distance and an analogue of the Koebe theorem for this class of mappings.