The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

Coupled connections on a compact Riemann surface

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K^⊗n_X⊗E, where K_X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.     

Value-sharing of meromorphic functions on a Riemann surface

We present some results on two meromorphic functions from S to sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane.     

Singular points on compactifications of a Riemann surface

A notion of capacity on general compactifications of a Riemann surface is introduced. It is used to establish the connection between singular points on the Royden compactification and singular points on the Kuramochi compactification. The second named author was supported by Grant-in-Aid for Scientific Research, Japanese Ministry of Education, Science and Culture.     

Remark on the size of a Riemann surface

Research partially supported by the National Science Foundation.     

An elliptic non-linear equation on a Riemann surface

New existence and uniqueness results for a second order elliptic non-linear equation are obtained by using gauge theory methods on linear holomorphic bundles over an oriented Riemann surface.     

Prym differentials on a variable compact Riemann surface

The dimensions and bases of the spaces of meromorphic Prym differentials on a variable compact Riemann surface, as well as in the first holomorphic de Rham cohomology group of Prym differentials, are found for any characters.     

Vector bundles on a nearly degenerate Riemann surface

We show that on a Riemann surface X close to a maximally degenerate complex curve, any semistable vector bundle of degree 0 is associated with a linear representation of the Schottky group ?? _X uniformizing X. Further, we study the relationship between the representation space of ?? _X and the moduli space of semistable bundles on X as a higher rank version of Abel?CJacobi??s theorem.     

Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff–Grothendieck bundle splitting \(\oplus _{i=1}^{r} \mathcal {O}(m_{i})\) over \(\mathbb {C}\mathbb {P}^{1}\) is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset \(S\subset \mathbb {C}\mathbb {P}^{1}\). The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types’ automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.     

Estimation for jump processes in the tangent bundle of a Riemann manifold

A stochastic process is formulated in the tangent bundle of a Riemann manifold where the vector fibre portion of the process is a jump process. Since the tangent spaces change as the process in the base manifold evolves, it is necessary to define a jump process in the fibres of the tangent bundle with respect to the process in the base manifold. An estimation problem is formulated and solved for a process obtained from the jump process in the fibres of the tangent bundle where the observations include the process in the base manifold and the jump times. Since each fibre of the tangent bundle is a linear space, a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.Research supported by NSF Grants ENG 75-06562 and MCS 76-01695 and AFOSR Grant 77-3177.     

Handle removal on a compact Riemann surface of genus 2

A degeneration of a compact, two-sheeted, Riemann surface in genus 2 is studied. Two branch points coalesce when the period matrix for the surface is degenerated. A Siegel modular transformation is applied to give the degenerating period matrix in the form of a corner matrix.     

On theta characteristics of a compact Riemann surface

Let σ be a nontrivial automorphism of a compact connected Riemann surface X of genus at least two. Assume that σ fixes each of the theta characteristics of X. We prove that X is hyperelliptic, and σ is the unique hyperelliptic involution of X.     

On automorphic factors of a compact Riemann surface

Summary Automorphic factors of degree n of a compact complex manifold are defined and the problem that ? Can every automorphic factor of degree n over a compact Riemann surface M be expressed as a product of a representation of the fundamental group of M and a certain canonical automorphic factor of M? ? is studied. The answer is found in affirmative for factors of degree1 and in negative for higher degree factors by explicit construction. Entrata in Redazione il 16 maggio 1972.     

Flows on a Riemann surface: a classification of flows on the torus

Developing the idea to use Riemann surfaces for modeling flat flows, we completely classify all flows that can be modeled using the Riemann surface of genus 1 (the torus).