Mathematische Zeitschrift

In this paper, we prove that every conformal minimal immersion of an open Riemann surface into $${\mathbb {R}}^n$$ for $$n\ge 5$$ can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into $${\mathbb {R}}^5$$ (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to $${\mathbb {R}}^n$$ for any $$n\ge 3$$ which is also proved in the paper (see Theorem 5.3).

We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable subgroup of local biholomorphisms has always recurrent orbits, i.e. there exists an orbit contained in its set of limit points.

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.

Arone and Lesh (J Reine Angew Math 604:73–136, 2007; Fund Math 207(1):29–70, 2010) constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg–MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh’s description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which was computed by Schwede (J Am Math Soc 30(3):673–711, 2017).