A Note on Cyclic Vectors for the Shift

We strengthen a result in the literature that gives a necessary condition for the shift on the Hardy space H 2 ( G ) of some bounded, simply connected region G to be cyclic. Part of this involves a characterization, in terms of ‘ G , of what it means for G to be the image of under a conformal map whose boundary values are univalent a.e. m (normalized Lebesgue measure on ).     

Continuity of the SLE trace in simply connected domains

We prove that the SLE _κ trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of harmonic measure.     

On the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups

The aim of this paper is to study the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups. We carry over a qualitative version of the Erdös-Rényi law of large numbers for Brownian motion to simply connected step 2-nilpotent Lie groups. The method applied gives rise to a proof for qualitative results concerning the modulus of continuity of Brownian motion on simply connected step 3-resp. step 2-nilpotent Lie groups without using the Ventsel-Freidlin theory as in Baldi.     

Carleson测度和多项式逼近

设G是复平面C上的单连通区域,在研究G上的Carleson测度和多项式在G上的Hardy空间Hq(G)中的稠密性问题时,建立起了两者之间的一种等价关系．用Carleson测度给出了多项式在Hq(G)中稠密的充要条件,也刻画了所有那些使得在其上的Carleson测度和Carleson不等式等价的单连通区域．     

Three-Term Recurrence Relation for Polynomials Orthogonal with Respect to Harmonic Measure

We prove that a three-term recurrence relation for analytic polynomials orthogonal with respect to harmonic measure in a simply connected domain G exists if and only if G is an ellipse.     

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.     

On an exterior boundary-value problem for the time-harmonic maxwell equations with boundary conditions for the normal components of the electric and magnetic field

We treat the time-harmonic Maxwell equations with the boundary condition (ν, E) = (ν, H) = 0 in an exterior multiply connected domain. A uniqueness result by Yee for the case of a simply connected domain is extended to multiply connected domains and existence is obtained by a boundary integral equation approach.     

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.     

Handle decompositions of simply connected five-manifolds. III

We prove the existence of exact handle decomposition of a simply connected smooth or piecewise linear (PL) five-manifold with a standard simply connected boundary of signature zero, the triviality of a five-dimensionalh-cobordism with ends of such type, and the uniqueness up to diffeomorphism (PL-isomorphism) of a smooth (PL)h-cobordism between a given simply connected four-manifold and the corresponding standard manifold.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 935–940, July, 1994.This work was partially supported by International Science Foundation, Grant No. U6F000.     

Tilting simply connected algebras

Let A be a representation-finite simply connected algebra, and T_Abe an arbitrary tilting module (in the sense of Happei and Ringel). We show that B = End T_A satisfies the separation condition of Bautista, Larrion and Salmeron. This implies that B is also simply connected.     

The lower density conjecture for harmonic measure

In this paper, we establish the lower density conjecture for harmonic measure in simply connected plane domains.     

On the bieberbach and Koebe constants for sector domains and sector disks

We discuss domain constants related to the classical Bieberbach and Koebe theorems. We find a class of simply connected domains for which the product of these constants behave like extremal domain and gives a better result on Osgood’s inequalities.     

Harmonic measure, infinite kernels, and symmetrization

The vanishing of area for the infinite Nielsen kernel of an arbitrary open Riemann surface is shown to follow from iteration of a natural geometric operation on the unit disk. This operation compares the distribution of harmonic measure on the boundaries of two related simply connected domains, and is not yet sufficiently well understood.

     

On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center-center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.     

On the equivariant formality of Kähler manifolds with torus group actions

We consider simply connected compact Kähler manifolds which have a holomorphic action of a torus group. We use the existing equivariant models for rational homotopy to show that these spaces satisfy an equivariant formality condition over the complex numbers.     

Einstein metrics and the number of smooth structures on a four-manifold

We prove that for every natural number k there are simply connected topological four-manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.     

Existence of non-isotropic conjugate points on rank one normal homogeneous spaces

We give a positive answer to the Chavel’s conjecture (J Differ Geom 4:13–20, 1970): a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is variational complete is a rank one symmetric space.     

A Helly-Type Theorem for Intersections of Compact Connected Sets in the Plane

For a family C of nonempty compact sets in the plane, the following conditions are equivalent:(1) Every two (not necessarily distinct) members of C have a connected union and every three (not necessarily distinct) members of C have a simply connected union.(2) C is a family of simply connected sets such that every two (not necessarily distinct) members of C have a connected intersection and every three (not necessarily distinct) members of C have a nonempty intersection.If either set of conditions is satisfied, then { C : C in C } is nonempty, simply connected, and connected. Furthermore, if the collection C is finite, then it is also true that { C : C in C } is simply connected.