Bilipschitz homogeneity and inner diameter distance

We prove that a Jordan plane domain whose boundary is bilipschitz homogeneous with respect to its inner diameter distance is a John disk. This opens the door to an abundance of equivalent conditions. We characterize such domains in terms of quasiconformal mappings as well as their Riemann maps. We introduce the notion of an inner diameter distance Jordan disk and present related results for these spaces.     

V-BILIPSCHITZ DETERMINACY OF WEIGHTED HOMOGENEOUS ANALYTIC FUNCTION-GERMS ON WEIGHTED HOMOGENEOUS REAL ANALYTIC VARIETIES

In this article, we provide estimates for the degree of V bilipschitz determinacy of weighted homogeneous function germs defined on weighted homogeneous analytic variety V satisfying a convenient Lojasiewicz condition.The result gives an explicit order such that the geometrical structure of a weighted homogeneous polynomial function germs is preserved after higher order perturbations.     

Evolution of conformal maps with quasiconformal extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS¹/RotS¹ embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space.     

Bilipschitz embedding of self-similar sets

In this paper, we prove that each self-similar set satisfying the strong separation condition can be bilipschitz embedded into each self-similar set with larger Hausdorff dimension. A bilipschitz embedding between two self-similar sets of the same Hausdorff dimension both satisfying the strong separation condition is only possible if the two sets are bilipschitz equivalent.     

Quasiconformal extension of quasimöbius mappings of Jordan domains

We introduce the new class of Jordan arcs (curves) of bounded rotation which includes all arcs (curves) of bounded turning. We prove that if the boundary of a Jordan domain has bounded rotation everywhere but possibly one singular point then every quasimöbius embedding of this domain extends to a quasiconformal automorphism of the entire plane.     

On Minimal Annuli with a Straight Line Boundary

Shiffman proved that if a minimal annulus A in a slab is bounded by two convex Jordan curves contained respectively in the two boundary planes P and Q of the slab, then A intersects all parallel planes between P and Q in strictly convex curves. We generalize Shiffman's result to the case that A is bounded by a strictly convex C² Jordan curve and a straight line. We show that in this case Shiffman's result is still true.     

Möbius bilipschitz homogeneous arcs on the plane

A möbius bilipschitz mapping is an η-quasimöbius mapping with the linear distortion function η(t) = Kt. We show that if an open Jordan arc γ ? C with distinct endpoints a and b is homogeneous with respect to the family F_K of möbius bilipschitz automorphisms of the sphere C with K specified then γ has bounded turning RT(γ) in the sense of Rickman and, consequently, γ is a quasiconformal image of a rectilinear segment. The homogeneity of γ with respect to F_K means that for all x, y ∈ γ {a, b} there exists f ∈ F_K with f(γ) = γ and f(x) = y. In order to estimate RT(γ) from above, we introduce the condition BR(δ) of bounded rotation of γ, and then the explicit bound depends only on K and δ.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Manifolds which are Ivanov-Petrova or k -Stanilov

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov.     

Digital Jordan curves

A digital Jordan curve theorem is proved for a new topology defined on Z². This topology is compared with the classical Khalimsky and Marcus topologies used in digital topology. We show that the Jordan curves with respect to the topology defined, unlike the Jordan curves with respect to any of the two classical topologies mentioned, may turn at the acute angle . We also discuss a quotient topology of the new topology.     

Homogeneous trees are bilipschitz equivalent

We prove that any two locally finite homogeneous trees with valency greater than 3 are bilipschitz equivalent. This implies that the quotienth ¹(G)/h ^k (G), whereh ^k (G) is thekthL ₂-Betti number ofG, is not a quasi-isometry invariant.     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L 1

We give sufficient conditions for a metric space to bilipschitz embed in L ₁. In particular, if X is a length space and there is a Lipschitz map ${u: X \rightarrow \mathbb R}$ such that for every interval ${I \subset \mathbb R}$ , the connected components of u ^?1(I) have diameter ${\leq {\rm const} \cdot {\rm diam}(I)}$ , then X admits a bilipschitz embedding in L ₁. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L ₁, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ? ₁ of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L ₁.     

Jordan Structures in Harmonic Functions and Fourier Algebras on Homogeneous Spaces

We study the Jordan structures and geometry of bounded matrix-valued harmonic functions on a homogeneous space and their analogue, the harmonic functionals, in the setting of Fourier algebras of homogeneous spaces.Supported by EPSRC grant GR/G91182 and NSERC grant 7679.     

Homogeneous spaces with invariant projectively flat affine connections

We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.

     

Embedding median algebras in products of trees

We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of $\mathbb{R }$ -trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete $\mathbb{R }$ -trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Dru?u and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of $\mathbb{R }$ -trees.     

Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric

In this paper existence of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric is established. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in C requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S¹). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales, together with a Douady-Earle type conformal extension of vector fields on the circle to the disk.     

The calculation of the L 2-norm of the index of a plane curve and related formulas

We provide formulas for calculating the L ²-norm of the index function of a rectifiable closed curve in the complex plane. Some applications to isoperimetric inequalities are given. The main tool used is the decomposition of a rectifiable closed curve into a sequence of Jordan curves, curves with null index functions, and an exceptional set.     

Pre-compactness of isospectral sets for the neumann operator on planar domains

The Neumann operator maps the boundary value of a harmonic function tc its normal derivative. The inverse spectral properties of the Neumann operator associated to smooth, planar, Jordan curves are studied. The Riemann mapping theorem is used tc parametrize the set of planar Jordan curves by positive functions on the unit circle. By studying the zeta function associated to the spectrum, it is shown that isospectral sets of these functions are pre-compact in the topology of the L²-Sobolev space of order 5/2 - [euro]. Spectral criteria are given for the limiting curves of an isospectral set to be Jordan. A spectrally determined lower bound on the area of the interior of the curve is given.