Universal Teichmüller space and BMO

We first give some new characterizations on BMOA–Teichmüller space and various characterizations on VMOA–Teichmüller space as well. In particular, we prove that a quasisymmetric conformal welding h$h$ corresponds to an asymptotically smooth curve in the sense of Pommerenke (1978) [32] precisely when h$h$ is absolutely continuous with logh^′∈VMO$log{h}^{\prime }\in \text{VMO}$. We then show that these BMO–Teichmüller spaces have natural complex structures.     

The Thurston boundary of Teichmüller space and complex of curves

Let S be a closed orientable surface with genus g?2. For a sequence σi in the Teichmüller space of S, which converges to a projective measured lamination [λ] in the Thurston boundary, we obtain a relation between λ and the geometric limit of pants decompositions whose lengths are uniformly bounded by a Bers constant L. We also show that this bounded pants decomposition is related to the Gromov boundary of complex of curves.     

Complex hyperbolic Kleinian groups with limit set a wild knot

We construct quasi-Fuchsian groups acting on two-dimensional complex hyperbolic space with limit set a wild knot. Also, we study the Teichmüller space T(G) of faithful, discrete, type-preserving representations of a Fuchsian group G of the first kind with parabolic elements in complex hyperbolic space. We show that T(G) is not connected, and that the Toledo invariant does not distinguish different connected components of T(G).     

Remarks on invariant metrics on Teichmüller space

We make some comparisons concerning the induced infinitesimal Kobayashi metric, the induced Siegel metric, the L² Bergman metric, the Teichmüller metric and the Weil-Petersson metric on the Teichmüller space of a compact Riemann surface of genus g?2. As a consequence, among others, we show that the moduli space has finite volume with respect to the L² Bergman metric. This answers a question raised by Nag in 1989.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

A Simpler Proof of Mirzakhani’s Simple Curve Asymptotics

Maryam Mirzakhani (in her doctoral dissertation) has proved the author’s conjecture that the number of simple closed curves of length bounded by L on a hyperbolic surface S is asymptotic to a constant times L^d, where d is the dimension of the Teichmüller space of S. In this note we clarify and simplify Mirzakhani’s argument.     

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.     

Moduli spaces of convex projective structures on surfaces

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL₃(R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.     

Real dicompactifications of ditopological texture spaces

Real dicompactifications and dicompactifications of a ditopological texture space are defined and studied.Section 2 considers nearly plain extensions of a ditopological texture space (S,S,τ,κ). Spaces that possess a nearly plain extension are shown to have a property, called here almost plainness, that is weaker than that of near plainness, but which shares with near plainness the existence of an associated plain space (Sp,Sp,τp,κp). Some properties of the class of almost plain ditopological texture spaces are established, a notion of canonical nearly plain extension of an almost plain ditopological texture space, projective and injective pre-orderings and the concept of isomorphism on such canonical nearly plain extensions are defined.In Section 3 the notion of nearly plain extension is specialized to that of real dicompactification and dicompactification, and the spaces that have such extensions are characterized. Working in terms of a specific representation of the canonical real dicompactifications and dicompactifications of a completely biregular bi-T₂ almost plain ditopological space, the interrelation between sub-T-lattices of the T-lattice of ω-preserving bicontinuous real mappings on the associated plain space and the real dicompactifications and dicompactifications are investigated. In particular generalizations of the Hewitt realcompactification and Stone-?ech compactification are obtained, and shown to be reflectors for the appropriate categories.     

Real structures of Teichmüller spaces

The moduli space X^g of compact Riemann surfaces of genus g, g>1, has a canonical antiholomorphic involution. It can easily be defined in terms of complex curves: a point in X^g represented by a curve C is mapped to the point represented by the complex conjugate ¯C of C. In other words, the moduli space has a canonical real structure (cf. Andreotti and Holm [2]). The Teichmüller space has, however, several essentially distinct real structures. The purpose of this note is to describe all real structures of the Teichmüller space T(g,n) of compact Riemann surfaces of genus g punctured at n points.Work supported by the EMIL AALTONEN FOUNDATION     

Holomorphic extensions of Laplacians and their determinants

The Teichmüller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det^′(Δ) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {Δμ,ν} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant of this family {Δμ,ν} and show how this family realizes the holomorphic extension of det^′(Δ) as its determinant.     

On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

Comparison between Teichmuller and Lipschitz metrics

We study the Lipschitz metric on a Teichmüller space (definedby Thurston) and compare it with the Teichmüller metric.We show that in the thin part of the Teichmüller spacethe Lipschitz metric is approximated up to a bounded additivedistortion by the sup-metric on a product of lower-dimensionalspaces (similar to the Teichmüller metric as shown by Minsky).In the thick part, we show that the two metrics are equal upto a bounded additive error. However, these metrics are notcomparable in general; we construct a sequence of pairs of pointsin the Teichmüller space, with distances that approachzero in the Lipschitz metric while they approach infinity inthe Teichmüller metric.     

Selective separability and its variations

A space X is said to be selectively separable (=M-separable) if for each sequence {Dn:n∈ω} of dense subsets of X, there are finite sets Fn⊂Dn (n∈ω) such that ?{Fn:n∈ω} is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X² is not selectively separable; (3) c{0,1} has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].     

Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation-projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, and with x∈R^d. For piecewise constant elements on the uniform grid of size n^d, we prove quadratic convergence O(h²) in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h³). A fast algorithm of complexity O(dR₁R₂nlogn) is described for tensor-product convolution on uniform/composite grids of size n^d, where R₁,R₂ are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O(h²) and O(h³); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n×n×n grid in the range n≤16384.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

Complex Symplectic Geometry of Quasi-Fuchsian Space

We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmüller space and is described in terms which look natural from the point of view of hyperbolic geometry.     

Kähler geometry of the universal Teichmüller space and coadjoint orbits of the Virasoro group

The Kähler geometry of the universal Teichmüller space and related infinite-dimensional Kähler manifolds is studied. The universal Teichmüller space T may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The classical Teichmüller spaces T(G), where G is a Fuchsian group, are contained in T as complex Kähler submanifolds. The homogeneous spaces Diff₊(S ¹)/Möb(S ¹) and Diff₊(S ¹)/S ¹ of the diffeomorphism group Diff₊(S ¹) of the unit circle are closely related to T. They are Kähler Frechet manifolds that can be realized as coadjoint orbits of the Virasoro group (and exhaust all coadjoint orbits of this group that have the Kähler structure).     

Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes

In the first part of the paper, we prove the existence of a unique global solution to the axisymmetric Navier-Stokes system with initial data and external force with . This improves the result obtained by S. Leonardi, J. Málek, J. Necǎs and M. Pokorný [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649], where H²(R³) regularity was required. In the second part, we state global existence and uniqueness for the axisymmetric Navier-Stokes system with initial data in W2,p(R³) and external force in with 1<p<2. This also improves [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649] because less integrability is required on v₀ and on f.     

Relations on the period mapping giving extensions of mixed hodge structures on compact Riemann surfaces

We consider a period map from Teichmüller space to , which is a real vector bundle over the Siegel upper half space. This map lifts the Torelli map. We study the action of the mapping class group on this period map. We show that the period map from Teichmüller space modulo the Johnson kernel is generically injective. We derive relations that the quadratic periods must satisfy. These identities are generalizations of the symmetry of the Riemann period matrix. Using these higher bilinear relations, we show that the period map factors through a translation of the subbundle and is completely determined by the purely holomorphic quadratic periods. We apply this result to strengthen some theorems in the literature. One application is that the quadratic periods, along with the abelian periods, determine a generic marked compact Riemann surface up to an element of the kernel of Johnson's homomorphism. Another application is that we compute the cocycle that exhibits the mapping class group modulo the Johnson kernel as an extension of the group SP _g () by the group .