Equivalence of closed 1-forms on surfaces with edge

We investigate closed 1-forms with isolated zeros on surfaces with edge. A criterion for the topological equivalence of closed 1-forms is proved.     

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

We study the new geometric flow that was introduced in the paper [12] of Topping and the author that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper of Topping and the author [12] where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.     

The minimal number of singularities of stable maps between surfaces

For a C^∞ stable map of a closed and connected surface M into the sphere, let c(φ), i(φ) and n(φ) denote the numbers of cusps, fold curves components and nodes respectively. In this paper, in a given homotopy class, we will determine the minimal pair (i,c+n) with respect to the lexicographic order.     

On quotient spaces of compact lie groups by Tori centralizers

We consider the graph of the homogeneous space K/L, where K is a compact Lie group and L is the centralizer of a torus in K. We obtain a characterization of those spaces whose graphs admit embeddings in a certain standard graph. We compute the number of arcs in such graphs. We also give a simple expression for the Euler class of the homogeneous space K/L.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

p-harmonic 1-forms on complete manifolds

Let (M ^m , g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L ^q (M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

Algebraic values of meromorphic functions on Riemann surfaces

We extend the Schneider-Lang theorem concerning algebraic values of meromorphic functions on the complex plane to meromorphic functions on punctured Riemann surfaces by using Jensen's formula for Riemann surfaces.     

Harmonic 1-forms on the stable foliation

We study cohomology classes of Hölder continuous closed leafwise 1-forms on the stable foliation of an Anosov geodesic flow. Each class contains a harmonic 1-form and is determined by its periods. Asymptotic quantities are computed in terms of the Pressure function defined by the geodesic flow.     

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.Received: 14 April 2004     

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE ⁰ can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin. Partially supported by a grant from NSERC of Canada.     

k-valent maps on the surfaces

In this paper, the works of Kotzig^[1], Zaks^[3] and the author^[2] have been generalized and unified. A number of new results have also been obtained.     

Second Gaussian maps on surfaces

In this article, we study second Gaussian maps on surfaces, both in general, and for specific examples. These examples include Hirzebruch surfaces and their double covers.