Surfaces Of Positive Genus Research Articles

Łojasiewicz inequalities for almost harmonic maps near simple bubble trees

We prove Łojasiewicz inequalities for the harmonic map energy for maps from surfaces of positive genus into general analytic target manifolds which are close to simple bubble trees and as a consequence obtain new results on the convergence of harmonic map flow and on the energy spectrum of harmonic maps with small energy. Our results and techniques are not restricted to particular targets or to integrable settings and we are able to lift general Łojasiewicz–Simon inequalities valid near harmonic maps \hat{\omega}\colon S^{2}\to N to the singular setting whenever the bubble \hat{\omega} is attached at a point which is not a branch point.

Outerspatial 2-Complexes: Extending the Class of Outerplanar Graphs to Three Dimensions

We introduce the class of outerspatial 2-complexes as the natural generalisation of the class of outerplanar graphs to three dimensions. Answering a question of O-joung Kwon, we prove that a locally 2-connected 2-complex is outerspatial if and only if it does not contain a surface of positive genus as a subcomplex and does not have a space minor that is a generalised cone over $K_4$ or $K_{2,3}$.
 This is applied to nested plane embeddings of graphs; that is, plane embeddings constrained by conditions placed on a set of cycles of the graph.

On the rigidity of Zoll magnetic systems on surfaces

In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level below the Mañe critical value. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.

Homological invariants of links in a thickened surface

In the paper we introduce a new invariant of oriented links in a thickened surface. The invariant is a generalization of polynomial invariants $$u_{\pm }(t)$$ introduced by Turaev in 2008. Our invariant $$Q(\ell )$$ is defined as follows. Consider a diagram representing an oriented link $$\ell \subset \varSigma \times [0,1]$$, where $$\varSigma $$ is a closed orientable surface of positive genus. A value of $$Q(\ell )$$ is the formal sum over all crossings in the diagram terms of the form $${\text {sign}}(c)[h_1(c),h_2(c)]$$, where $${\text {sign}}(c)$$ denotes the sign of a crossing c and $$[h_1(c),h_2(c)]$$ denotes a ordered pair of homology classes of two loops associated with the crossing. As an application we prove a low bounds for the crossing number and for the virtual genus of a link. Additionally we describe an analogous constructions in some other situations, in particular, in the case of long virtual knots and prove a low bound for the virtual genus of a virtual knot which can be represented by concatenation of long virtual knots. Finally we show that Turaev’s invariants $$u_{\pm }(t)$$ is weaker than $$Q(\ell )$$ and discuss the results of a computing experiment which illustrates the fact.

Generalizations of hyperbolic area for topological surfaces

We introduce two generalizations of hyperbolic area for connected, closed, orientable surfaces: the complexity and the simple complexity of a surface. These concepts are defined in terms of collections of branched coverings M→P1, where M is a Riemann surface homeomorphic to S and P1 is the Riemann sphere. We prove that if S is a surface of positive genus, then both the topological complexity and the simple topological complexity of S are linear functions of its genus.

Modular flip-graphs of one-holed surfaces

We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus g with a single boundary curve and n marked points on this curve and consider triangulations up to homeomorphism with the marked points as their vertices. Our results are bounds on the maximal distance between two triangulations. Our lower bounds assert that these distances grow at least like 5n∕2 for all g≥1. Our upper bounds grow at most like [4−1∕(4g)]n for g≥2, and at most like 23n∕8 for the bordered torus.

Smooth long-time existence of Harmonic Ricci Flow on surfaces

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.

Dehn’s algorithm for simple diagrams

Dehn gave an algorithm for deciding if two cyclic words in the standard presentation of the fundamental group of a closed oriented surface of positive genus represent the same conjugacy class. A simple diagram on a surface is a disjoint union of simple closed curves none of which bound a disk. If [Formula: see text] is a once punctured closed surface of negative Euler characteristic, simple diagrams are classified up to isotopy by their geometric intersection numbers with the edges of an ideal triangulation of [Formula: see text]. Simple diagrams on the unpunctured surface [Formula: see text] can be represented by simple diagrams on [Formula: see text]. The weight of a simple diagram is the sum of its geometric intersection numbers with the edges of the triangulation. We show that you can pass from any representative to a least weight representative via a sequence of elementary moves, that monotonically decrease weights. This leads to a geometric analog of Dehn’s algorithm for simple diagrams.

Chaotic Expansive Homeomorphisms on Closed Orientable Surfaces of Positive Genus

In this paper we give a new and elementary proof to the following fact: each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism. Further more, we show that the homeomorphisms given are also weakly mixing.

Embedded graphs whose links have the largest possible number of components

We derive the basic properties of graphs embedded on surfaces of positive genus whose corresponding link diagrams have the largest possible number of components.

Kodaira dimension and symplectic sums

Modulo trivial exceptions, we show that symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T2-bundles over T2 each having b1 = 2.

Small exotic 4–manifolds

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T^2-bundles over T^2 each having b_1=2.

Virtually Haken fillings and semi-bundles

Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along dM are virtually Haken. It follows that infinitely many Dehn-surgeries of any non-trivial knot in the three-sphere are virtually Haken.

Bounded cohomology and non-uniform perfection of mapping class groups

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective.

Three-Coloring Graphs Embedded on Surfaces with All Faces Even-Sided

Every graph embedded on a surface of positive genus with every face bounded by an even number of edges can be 3-colored provided all noncontractible cycles in the graph are sufficiently long. The bound of three colors is the smallest possible for this type of result.

Automorphism properties of embedded graphs

AbstractThis paper presents some conditions under which every automorphism of a graph, embedded on a surface of positive genus, maps face boundaries of the embedding to face boundaries. These results extend a result of Whitney which provides an analogous, but simpler, condition for planar graphs. The new result is applied to graphs on the torus to obtain a classification of many toroidal vertex‐, edge‐, and face‐transitive graphs.

Arcbodies

A Haken manifold is a compact, orientable, irreducible 3-manifold which contains a properly embedded 2-sided, incompressible surface of positive genus. These manifolds are important in connection with the work of Haken, Waldhausen and the more recent work of Thurston (8). Thus it is interesting to investigate criteria for testing incompressible surfaces on 3-manifolds.

Abstract loop equations, topological recursion and new applications

loop equations, topological recursion and applications Gaetan Borot, Bertrand Eynard, Nicolas Orantin Abstract We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.