POLYNOMIAL HULLS AND ENVELOPES OF HOLOMORPHY OF SUBSETS OF STRICTLY PSEUDOCONVEX BOUNDARIES

First we give a construction of a Cantor set in the unit sphere in ℂ3 the polynomial hull of which contains interior points. Such sets were known to exist in spheres in ℂ2. Secondly, we construct an open connected subset of the unit sphere in ℂ3 with infinitely sheeted envelope of holomorphy.

ON RIGID AND INFINITESIMAL RIGID DISPLACEMENTS IN THREE-DIMENSIONAL ELASTICITY

Let Ω be an open connected subset of ℝ3 and let Θ be an immersion from Ω into ℝ3. It is first established that the set formed by all rigid displacements, i.e. that preserve the metric, of the open set Θ(Ω) is a submanifold of dimension 6 and of class [Formula: see text] of the space H1(Ω). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the same set Θ(Ω) is nothing but the tangent space at the origin to this submanifold. In this fashion, the familiar "infinitesimal rigid displacement lemma" of three-dimensional linearized elasticity is put in its proper perspective.

A CHARACTERIZATION OF CONVEXITY-PRESERVING MAPS FROM A SUBSET OF A VECTOR SPACE INTO ANOTHER VECTOR SPACE

Let V and X be Hausdorff, locally convex, real, topological vector spaces with dim V > 1. It is shown that a map σ from an open, connected subset of V onto an open subset of X is homeomorphic and convexity-preserving if and only if σ is projective.

Local homeo- and diffeomorphisms: invertibility and convex image

We prove a necessary and sufficient condition for a local homeomorphism defined on an open, connected subset of a Euclidean space to be globally one-to-one and, at the same time, for the image to be convex. Among the applications we give a practical sufficiency test for invertibility for twice differentiable local diffeomorphisms defined on a ball.

A new proof of the Brouwer plane translation theorem

Let f be an orientation-preserving homeomorphism of ℝ2 which is fixed point free. The Brouwer ‘plane translation theorem’ asserts that every x0 ∈ ℝ2 is contained in a domain of translation for f i.e. an open connected subset of ℝ2 whose boundary is L ∪ f(L) where L is the image of a proper embedding of ℝ in ℝ2, such that L separates f(L) and f−1(L). In addition to a short new proof of this result we show that there exists a smooth Morse function g: ℝ2 → ℝ such that g(f(x)) < g(x) for all x and the level set of g containing x0 is connected and non-compact (and hence the image of a properly embedded line).

An estimate of harmonic measure in Ṟd, d ≧ 2

SynopsisLet D be an open connected subset of the open unit ball in Ṟd, d ≧ 2. We give an estimate of the harmonic measure of ∂D∩{|x| = 1} with respect to D. This estimate depends in a simple way on the geometry of D. An essential tool is a rearrangement theorem for differential inequalities. When d = 2, examples are given which illustrate the precision of the results.