On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

Conformal uniformization and packings

A new short proof is given for Brandt and Harrington’s theorem about conformal uniformizations of planar finitely connected domains as domains with boundary components of specified shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of specified shapes and with specified combinatorics. The shapes can be arbitrary smooth disks specified up to homothety, for example. The combinatorics of the packing is described by the contacts graph, which can be specified to be any finite planar graph whose vertices correspond to the shapes. This is in the spirit of Koebe’s proof of the Circle Packing Theorem as a consequence of his uniformization by circle domains. The author thankfully acknowledges support of NSF grant DMS-9112150.     

Uniform,Sobolev extension and quasiconformal circle domains

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

Phase space,wavelet transform and Toeplitz-Hankel type operators

Domain constants are numbers attached to regions in the complex plane ℂ. For a region Ω in ℂ, letd(Ω) denote a generic domain constant. If there is an absolute constantM such thatM ⁻¹≤d(Ω)/d(Δ)≤M whenever Ω and Δ are conformally equivalent, then the domain constant is called quasiinvariant under conformal mappings. IfM=1, the domain constant is conformally invariant. There are several standard problems to consider for domain constants. One is to obtain relationships among different domain constants. Another is to determine whether a given domain constant is conformally invariant or quasi-invariant. In the latter case one would like to determine the best bound for quasi-invariance. We also consider a third type of result. For certain domain constants we show there is an absolute constantN such that |d(Ω)−d(Δ)|≤N whenever Ω and Δ and conformally equivalent, sometimes determing the best possible constantN. This distortion inequality is often stronger than quasi-invariance. We establish results of this type for six domain constants. Research partially supported by a National Science Foundation Grant.     

Fridman’s Invariant,Squeezing Functions,and Exhausting Domains

We show that if a bounded domain Ω is exhausted by a bounded strictly pseudoconvex domain D with C2 boundary, then Ω is holomorphically equivalent to D or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman’s invariant has certain growth condition near the boundary.     

Pseudo Jordan domains and reflecting Brownian motions

Summary The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC ¹ curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when G has finite 1-dimensional lower Minkowski content.     

The planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.     

Metrics of Hyperbolic Type on Bounded Fractals

On the bounded Sierpinski gasket F we use the set of essential fixed points V ₀ as a boundary and consider the fractal Brownian motion on F killed in V ₀. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the local energy of harmonic functions. Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’     

Boundary conforming Delaunay mesh generation

A boundary conforming Delaunay mesh is a partitioning of a polyhedral domain into Delaunay simplices such that all boundary simplices satisfy the generalized Gabriel property. It’s dual is a Voronoi partition of the same domain which is preferable for Voronoi-box based finite volume schemes. For arbitrary 2D polygonal regions, such meshes can be generated in optimal time and size. For arbitrary 3D polyhedral domains, however, this problem remains a challenge. The main contribution of this paper is to show that boundary conforming Delaunay meshes for 3D polyhedral domains can be generated efficiently when the smallest input angle of the domain is bounded by arccos 1/3 ≈ 70.53°. In addition, well-shaped tetrahedra and an appropriate mesh size can be obtained. Our new results are achieved by reanalyzing a classical Delaunay refinement algorithm. Note that our theoretical guarantee on the input angle (70.53°) is still too strong for many practical situations. We further discuss variants of the algorithm to relax the input angle restriction and to improve the mesh quality.     

On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in \mathbbR^N{\mathbb{R}^N} of given volume. When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant in the boundary condition, or equivalently of the domain’s volume. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions.     

A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

Nondegeneracy of elliptic problems with periodic nonlinearities in the plane

A class of asymptotically quadratic functionals on Hilbert spaces, called degenerate, is considered and explored. Our results are applied to obtain a extension, in the planar case, of a result published by Solimini in ‘On the solvability of some elliptic partial differential equations with the linear part at resonance’, J. Math. Anal. Appl., 117 (1986), 138-152. Similar extensions had been previously studied in the literature only for domains with particular geometries.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

The conformal ‘bratwurst’ mapsand associated Faber polynomials

Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of ‘bratwurst’ shape sets in the complex plane are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A ‘bratwurst’ shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis onto the unit circle. Such sets are well suited to serve as inclusion sets for sets associated with a matrix, for example the spectrum, field of values or a pseudospectrum. Hence, the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems. The main advantage of the approach is that the conformal maps are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic convergence factor and Faber polynomials. Numerical examples are given. Received October 7, 1998 / Revised version received March 15, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in \mathbbC²{\mathbb{C}^{2}} near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range—roughly speaking—from being “mildly infinite-type” to very flat at the infinite-type points.     

Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C ^∞-convergence on compact subsets.      

On the role of convexity in isoperimetry,spectral gap and concentration

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-émery. Supported by NSF under agreement #DMS-0635607.     

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

We show how ‘test’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. Received: 13 June 2005