Teichmüller Theory of Bordered Surfaces

We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ k on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index k of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues σ k are uniformly bounded in k .

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

On the existence of heterotic-string and type-II-superstring field theory vertices

We study decorated mapping class groups, i.e., mapping class groups of surfaces with marked points and boundary components, and their behaviour under stabilization maps with respect to the genus, the number of punctures and boundary components. Decorated mapping class groups are non-trivial extensions of the undecorated mapping class group, and the first result states that the extension is homologically trivial when one stabilizes with respect to the genus. The second result implies that one also gets splittings of homology groups when stabilizing with respect to the number of punctures and boundary components.

Let 5 be a compact bordered Riemann surface of genus g with k boundary components. If 2g-\-k—1^2, the automorphism group of S is a finite group. Then, we put N(g, k) the maximum order of automorphism groups of S where the maximum is taken over all S of genus g with k boundary components. It is well known that N(g, k) is equal to the maximum order of automorphism groups of compact Riemann surfaces of genus g deleted k points, and every automorphism group of S is isomorphic to that of a compact Riemann surface (Oikawa [7]). For every k^O, N(0, k), N(l, k), N{2, k), N(3, k) and 7V(4, k) are determined by Heins [2], Oikawa [7], Tsuji [8], Tsuji [9] and Kato [4], respectively. In the present paper, we shall determine N(5, k). Theorem. JV(5, k) is

We start by pointing out that certain Riemann surfaces appear\nrather naturally in the context of wave equations in the black hole background.\nFor a given black hole there are two closely related surfaces.\nOne is the Riemann surface of complexified "tortoise" coordinate. The\nother Riemann surface appears when the radial wave equation is interpreted\nas the Fuchsian differential equation. We study these surfaces\nin detail for the BTZ and Schwarzschild black holes in four and higher\ndimensions. Topologically, in all cases both surfaces are a sphere with a\nset of marked points; for BTZ and 4D Schwarzschild black holes there\nis 3 marked points. In certain limits the surfaces can be characterized\nvery explicitly. We then show how properties of the wave equation\n(quasi-normal modes) in such limits are encoded in the geometry of the\ncorresponding surfaces. In particular, for the Schwarzschild black hole\nin the high damping limit we describe the Riemann surface in question\nand use this to derive the quasi-normal mode frequencies with the log 3\nas the real part. We then argue that the surfaces one finds this way signal\nan appearance of an effective string. We propose that a description\nof this effective string propagating in the black hole background can be\ngiven in terms of the Liouville theory living on the corresponding Riemann\nsurface. We give such a stringy description for the Schwarzschild\nblack hole in the limit of high damping and show that the quasi-normal\nmodes emerge naturally as the poles in 3-point correlation function in\nthe effective conformal theory.

It is known that the partially ordered set of all tuples of pairwise non- intersecting diagonals in an -gon is isomorphic to the face lattice of a convex polytope called the associahedron. We replace the -gon (viewed as a disc with marked points on the boundary) by an arbitrary oriented surface with a set of labelled marked points (`vertices'). After appropriate definitions we arrive at a cell complex (generalizing the associahedron) with the barycentric subdivision . When the surface is closed, the complex (as well as ) is homotopy equivalent to the space of metric ribbon graphs or, equivalently, to the decorated moduli space . For bordered surfaces we prove the following. 1) Contraction of an edge does not change the homotopy type of the complex. 2) Contraction of a boundary component to a new marked point yields a forgetful map between two diagonal complexes which is homotopy equivalent to the Kontsevich tautological circle bundle. Thus we obtain a natural simplicial model for the tautological bundle. As an application, we compute the psi- class, that is, the first Chern class in combinatorial terms. This result is obtained by using a local combinatorial formula. 3) In the same way, contraction of several boundary components corresponds to the Whitney sum of tautological bundles.

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a pair. A pair is the relation algebraic analogue of a relation of the form {(a, a), (b, b)} (with a = b allowed). In a simple pair-dense relation algebra, every pair is either a point (an algebraic analogue of {(a, a)}) or a twin (a pair which contains no point). In fact, every simple pair-dense relation algebra 2( is completely representable over a set U iS | Ul = K + 2R, where K iS the number of points of 2( and A is the number of twins of 2( . A relation algebra is said to be point-dense if every nonzero element below the identity contains a point. In a point-dense relation algebra every pair is a point, so a simple point-dense relation algebra 2( is completely representable over U iS IUI = c, where K iS the number of points of 2( . This last result actually holds for semiassociative relation algebras, a class of algebras strictly containing the class of relation algebras. It follows that the relation algebra of all binary relations on a set U may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as U . Semiassociative relation algebras may not be associative, so the equation (x; y); z = x; (y; z) may fail, but it does hold if any one of x, y, or z is 1. In fact, any rearrangement of parentheses is possible in a term of the form x0; ...; X(X_1 X in case one of the XKSS is 1. This result is proved in a general setting for a special class of groupoids.

A finite bordered Riemann surface 9-with s boundary components and interior genus g has first Berti number r = 2g + s -1.Let a be any interior point of 9, and e" denote evaluation at a on the usual hypo-Dirichlet algebra associated with 9. We establish some connections between the topological and, more strongly, the conformai type of 9, and the geometry of 91t" the set of representing measures for e".For example, we show that if 91t" has an isolated extreme point, then ft must be a planar surface.Several questions posed by Sarason are answered through exhausting the possibilities for the case r = 2.1. Let fi be a finite open Riemann surface with boundary oR consisting of a finite nonzero number s of disjoint Jordan curves (see Ahlfors and Sario [4, p. 117]).The first Betti number of R, which we denote by r, equals 2g + s -1, where g is the interior genus of R. We take 9fi to be oriented with the induced orientation from R. Lei A be the hypo-Dirichlet algebra (see [2], [15]) of functions continuous on oR and possessing continuous extensions to T = R U oR which are holomorphic on R. Fix a point a E R and let ea denote evaluation on A at a; i.e., ea(f) = f(a) for/ G A. Let 9ILa be the compact convex set of representing measures for ea.A measure m E 91ta is a positive Borel measure of total mass one supported on 3fi such that/(a) = Smfdm for all/ G A.In this paper we shall show some connections between the topological and, more strongly, the conformai type of R and the geometry of 91t,,.We will let S2 denote the Riemann sphere and Re be the real axis.The set <Dlta will be called strictly convex if 991ta consists entirely of extreme points.The principal results are stated below.Theorem 1.1.If 911,, has an isolated extreme point, then R is conformally equivalent to S2 minus a finite number of slits contained in Re.The next theorem exhausts the possibilities for 91ta when r = 2, and thus answers Questions 1 and 3 on p. 376 of [12].

In the perturbative formulation of string theory the partition function in the background of D-branes can be expanded as $$Z_{string} = \sum\limits_{g,b} {g_s^{ - 2 + 2g + b} Z_{g,b} .}$$ (1) Here Z g,b is the value of the diagram given by a Riemann surface of genus g with b boundary components. One way to compute the numbers Z g,b is to work in conformai gauge. In this gauge one obtains a conformai field theory (CFT) living on the string world sheet, and the Z g,b result from integrating correlators of this CFT over the moduli space of the Riemann surface. In order for this approach to make sense, the CFT has to be well defined on surfaces of arbitrary genus and with an arbitrary number of boundaries. For type I strings also non-orientable surfaces have to be included in the expansion (1). In this note we restrict ourselves to type II strings and orientable world sheets.

This paper provides a complete classification of four-dimensional complex Poisson algebras, extending previous work on lower-dimensional cases. We develop a systematic approach that combines the classification of four-dimensional complex commutative associative algebras with the classification of four-dimensional complex Lie algebras, allowing us to determine all compatible Poisson structures up to isomorphism. Our main result presents 71 distinct types: 60 isolated non-isomorphic Poisson algebras with non-trivial commutative product and 11 one-parameter families of non-isomorphic Poisson algebras. In addition to these, we include the known four-dimensional complex Lie algebras, viewed as Poisson algebras with trivial commutative product. This purely algebraic classification provides a foundation for future studies, including the geometric classification of Poisson algebras, their degenerations, and the analysis of orbit closures in the corresponding algebraic variety.

Let f : M→R 2 be a smooth immersion of a compact connected oriented surface with boundary M into R 2 . Kauffman defined an equivalence relation called image homotopy and classified the set of all orientation preserving immersions of M into R 2 up to image homotopy. When M is of genus one and the number of boundary components is strictly greater than one, Kauffman's result requires a correction. In this paper we will study this particular case.

: A minimal surface is the surface of least area bounded by a given closed curve. In three dimensional space these surface are realized, for reasonably simple curves, by soap films spanning wire loops. This paper obtains results related to the theorem that the normals to a complete minimal surface, not a plane, are everywhere dense. Examples: l. The same theorem holds for two-dimensional surfaces in Euclidean spaces of dimension three or more. 2. Theorems connecting total curvature, Euler characteristic, and number of boundary components. 3. Theorems about the normals and tangent planes to minimal surfaces, including the capacity of the set of directions omitted. (Author)

Biological systems often rely on topological transformation to reconfigure connectivity between nodes to guide the flux of molecular information. Here we develop a topology-programmed DNA origami system that encodes signal propagation at the nanoscale, analogous to topologically efficient information processing in cellular systems. We present a systematic molecular implementation of topological operations involving 'glue-cut' processes that can prompt global conformational change of DNA origami structures, with demonstrated major topological properties including genus, number of boundary components and orientability. By spatially arranging reactive DNA hairpins, we demonstrate signal propagation across transmission paths of varying lengths and orientations, and curvatures on the curved surfaces of three-dimensional origamis. These DNA origamis can also form dynamic scaffolds for regulating the spatial and temporal signal propagations whereby topological transformations spontaneously alter the location of nodes and boundary of signal propagation network. We anticipate that our strategy for topological operations will provide a general route to manufacture dynamic DNA origami nanostructures capable of performing global structural transformations under programmable control.

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = {\pi}_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .