MathSciDoc: An Archive for Mathematician ∫

We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for$n$>1 which contain a given compact set$K$and at the same time avoid a totally real affine subspace$L$of dimension <$n$, provided that$K$∪$L$is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤$k$<$n$enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension <$n$.

We characterize mappings$S$_{$i$}and$T$_{$i$}, not necessarily linear, from sets $\mathcal {J}_{i}$ ,$i$=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products:$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))⊂$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$)) and$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))∩$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$))≠∅, $a\in \mathcal {J}_{1}$ , $b\in \mathcal {J}_{2}$ . As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

We show that the standard graded cover of the well-known category $\mathcal{O}$ of Bernstein–Gelfand–Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ^{$h$}as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ^{$h$}. Finally we define what it means for$h$to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.

For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.

We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on $\mathbb{R}^{2}$ with the generalized homogeneity.

We show that positive$α$-stable densities are hyperbolically completely monotone if and only if$α$≤1/2. This gives a positive answer to a question raised by L. Bondesson in 1977.

We show that there exist $(d-1)$ -Ahlfors regular compact sets $E \subset\mathbb{R}^{d}$ , $d\geq2$ such that for any $t< d-1$ , we have $$\sup_{T} \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^{t}}< \infty $$ where the supremum is over all tubes $T$ with width $w(T) >0$ . This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.

Given a family of nodal curves, a semistable modification of it is another family made up of curves obtained by inserting chains of rational curves of any given length at certain nodes of certain curves of the original family. We give comparison theorems between torsion-free, rank-1 sheaves in the former family and invertible sheaves in the latter. We apply them to show that there are functorial isomorphisms between the compactifications of relative Jacobians of families of nodal curves constructed through Caporaso’s approach and those constructed through Pandharipande’s approach.