Let X be a compact quotient of the product of the real Heisenberg group H _4m+1 of dimension 4m + 1 and the three-dimensional real Euclidean space R ³ . A left-invariant hypercomplex structure on H _4m+1 R ³ descends onto the compact quotient X. The space X is a hyperholomorphic fibration of 4-tori over a 4m-torus. We calculate the parameter space and obstructions to deformations of this hypercomplex structure on X. Using our calculations, we show that all small deformations generate invariant hypercomplex structures on X but not all of them arise from deformations of the lattice. This is in contrast to the deformations on the 4m-torus.

Rigidity question have attracted much interest in the past. In the compact case, we have the famous work of Calabi and Vesentini [3] and Mostow [17]. Whereas Calabi and Vesentini proved a local version, namely that compact quotients of bounded symmetric domains admit no nontrivial deformations in case the domain is irreducible and of complex dimension at least 2, Mostow proved a global rigidity result, at the expense, however, of working only within the class of quotients of symmetric domains. Mostow's work is based on quasiconformal mappings. A different analytic approach was recently undertaken by Siu [22]. If M is a compact K~ ihler manifold diffeomorphic (or, more generally, homotopically equivalent) to a quotient N of an irreducible bounded symmetric domain, he studied a harmonic homotopy equivalence the existence of which is assured by the theorem of EeUs and Sampson, and demonstrated that

Let <i>L</i> be a holomorphic line bundle over a compact Khler manifold <i>X</i>. Motivated by mirror symmetry, we study the deformed HermitianYangMills equation on <i>L</i>, which is the line bundle analogue of the special Lagrangian equation in the case that <i>X</i> is CalabiYau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that <i>X</i> is a Khler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when <i>L</i> is ample and <i>X</i> has non-negative orthogonal bisectional curvature.

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set X in C3 whose polynomial hull is strictly larger than X but contains no analytic discs.

Let ψ be a conformal map on D with ψ(0)=0 and let F_α={z∈D:|ψ(z)|=α} for α>0. Denote by H^p(D) the classical Hardy space with exponent p>0 and by h(ψ) the Hardy number of ψ. Consider the limits L:=lim_{α→+∞} (log ω_D(0,Fα)^{−1} / logα), μ:=lim_{α→+∞}(d_D(0,Fα) / logα), where ω_D(0,Fα) denotes the harmonic measure at 0 of F_α and d_D(0,Fα) denotes the hyperbolic distance between 0 and F_α in D. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and h(ψ)? Motivated by the result of Kim and Sugawa that h(ψ)=lim inf_{α→+∞} (log ω_D(0,Fα)^{−1} logα), we show that h(ψ)=lim inf_{α→+∞} (d_D(0,Fα) / logα). We also provide conditions for the existence of L and μ and for the equalities L=μ=h(ψ). Poggi-Corradini proved that ψ∉H^μ(D) for a wide class of conformal maps ψ. We present an example of ψ such that ψ∈H^μ(D).