Normalized Ricci flow on nonparabolic surfaces

This paper studies the normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically –1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature –1. A relative estimate of Green’s function is proved as a tool.      

Ricci solitons on compact Kähler surfaces

We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.

     

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.

     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.      

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.     

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.

     

On Hermitian surfaces with J-invariant Ricci tensor

We prove that a compact Hermitian surface with J-invariant Ricci tensor is K?hler provided that the difference of its scalar and conformal scalar curvature is constant. In particular, there are no locally homogeneous examples of such surfaces with odd first Betti number. Received 20 July 2000.     

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.     

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose is a connected Riemann surface. Let denote the homeomorphism group of with the compact-open topology, and denote the subgroup of quasiconformal mappings of onto itself, and let and denote the identity components of and respectively. In this paper we show that the pair is an -manifold, and determine their topological types.

     

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.     

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption, we derive a formula for the derivative of Yamabe constant , where g(t) is a solution of Ricci flow on closed manifold. We also give a simple application. Mathematics Subject Classifications (2000): 53C21 and 53C44     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

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.

     

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

We study the injectivity radius bound for 3-d Ricci flow with bounded curvature. As applications, we show the long time existence of the Ricci flow with positive Ricci curvature and with curvature decay condition at infinity. We partially settle a question of Chow-Lu-Ni [Hamilton’s Ricci Flow, p. 302].     

A simple proof on the non-existence of shrinking breathers for the Ricci flow

Suppose M is a compact n-dimensional manifold, n≥ 2, with a metric g _ij (x, t) that evolves by the Ricci flow ∂ _t g _ij = −2R _ij in M× (0, T). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. Mathematics Subject Classification (1991) Primary 58J35, 53C44 Secondary 58C99