Solving Beltrami equations by circle packing

In Theorem 1 and Proposition 2 we generalize these results of Thurston to com­ pact bordered triangulated surfaces. Our proof relies on Thurston's result for closed surfaces. It follows the lines of Perron's method for solving the Dirichlet Problem using subharmonic functions. In Perron's method, the harmonic solu­ tion is the upper envelope of a family of subharmonic functions. To actually pro­ duce the harmonic solution one must exhibit suitable subharmonic functions (barriers) in the Perron family. The circle packing case is analogous: we obtain the solution as the upper envelope of a Perron-type family. Nonsingular circle packings are the analogues of harmonic functions, and circle packings with neg­ ative curvature cone type singularities are the analogues of subharmonic functions. We use Thurston's result for closed surfaces in order to produce a negative cur­ vature circle packing and thereby show that the family is nontrivial. The bound­ ary value problem and its solution by Perron family techniques were introduced by Peter Doyle [8] and Carter and Rodin [6,7]. They treated the euclidean case for genus :51; Beardon-Stephenson [4] treated the hyperbolic simply connected case by a related method. For the results above we need a suitable notion of "circle packing on a bordered surface" when circles on the border are to have infinite hyperbolic radius. There are two obvious possibilities for such a notion: infinite border circles can be re­ quired to be tangent to the border or they can be required to be orthogonal to the border. That is, a hyperbolic circle of infinite radius might be defined to be ei­ ther a horocycle or a hyperbolic geodesic. With either choice one can define the

Harmonic maps between pseudo-Riemannian surfaces

We consider the so-called G-systems for functions with values in a Lie group G which are generalizations of regular systems of ordinary linear differential equations. The Riemann-Hilbert monodromy problem is studied in detail together with the Riemann-Hilbert nonlinear boundary problem. We also establish some properties of the moduli space of complex structures on a principal bundle over a Riemann surface and discuss their connections with the Beltrami equation and generalized analytic functions.

We prove that the quasiregular mappings given by the (normalized) principal solution of the linear Beltrami equation (1) and the principal solution of the quasilinear Beltrami equation are inverse to each other. This basic fact is deduced from the Liouville for generalized analytic functions. It essentially simplifies the known proofs of the measurable Riemann mapping theorem and its holomorphic dependence on parameters.

CHAPTER II - REDUCTION OF A POSITIVE DIFFERENTIAL QUADRATIC FORM TO THE CANONICAL FORM. BELTRAMI'S EQUATION. GEOMETRIC APPLICATIONS

We draw connections between the field of contact topology (the study of totally non-integrable plane distributions) and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields (non-zero fields parallel to their non-zero curl). This immediately yields existence proofs for smooth, steady, fixed-point free solutions to the Euler equations on all 3-manifolds and all subdomains of 3 with torus boundaries. This correspondence yields a hydrodynamical reformulation of the Weinstein conjecture from symplectic topology, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all rotational Beltrami flows on S 3. This is the key step for a positive solution to a `hydrodynamical' Seifert conjecture: all steady flows of a perfect incompressible fluid on S 3 possess closed flowlines. In the case of spatially periodic Euler flows on 3, we give general conditions for closed flowlines derived from the algebraic topology of the vector field.

We use branched circle packings to construct an approximating sequence to a proper quasiregular mapping, that is a general proper solution of a Beltrami equation. We show that the approximating sequence converges on compact subsets, and show that for given branch-set the solution is unique subject to a certain normalization.

The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.

Given a Riemann surface S, there exists a finitely generated Fuchsian group G of the first kind acting on the upper half plane U, such that S≅U/G. This isomorphism makes it possible to use Fuchsian group methods to prove theorems about Riemann surfaces. In this note we give a proof of the Serre duality theorem by Fuchsian group methods which is technically simpler than proofs depending on sheaf theoretic methods.

Boundary value problems and periodic solutions for contingent differential equations

We consider the Laplace–Beltrami operator on a compact Riemann surface of a constant negative curvature. For any eigenvalue of the Laplace–Beltrami operator there is an associated sequence of measures on the Riemann surface. These measures naturally appear in Quantum Chaos type questions in the theory of electro-magnetic flow on a Riemann surface. The main result of the paper is the claim that this sequence of measures has the Liouville measure as the (weak*) limit. We prove a quantitative version of this equidistribution claim.

In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [12] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case.We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O(√g/n) and vertex bisectors of size O(√gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces.

The construction of geometric surfaces via labelled complexes was introduced by Thurston[16, chapter 13], and subsequent applications and developments have appeared in [1, 3, 4, 5, 14, 15]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[1] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [2] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [3, 4] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [5] and Doyle [8] developed the method for flat surfaces and Minda and Rodin [14] developed the method for finite type surfaces. Minda and Rodin [14] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph $G$. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice $A_n$ analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of $A_n$. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.

The articles in these two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. This first volume contains contributions from many of the world's leading figures in this field, and their contributions demonstrate the many interesting facets of geometrical group theory. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library