On the asymptotic Plateau problem for cycles in the Tits boundary of a Hadamard space

Abstract

The dimension of cycles in the Tits boundary of a proper Hadamard space is bounded by the asymptotic rank m of the space minus one. Kleiner and Lang proved that for (m-1)-dimensional cycles in the Tits boundary, the asymptotic Plateau problem can be solved. We prove that the asymptotic Plateau problem can be solved for so called strongly immovable cycles in the Tits boundary of a proper Hadamard space. The class of strongly immovable cycles contains all (m-1)-cycles. Further, it is a result of Huang, Kleiner and Stadler that the class of strongly immovable cycles contains the boundaries of cocompact flats which do not bound flat half-spaces and we prove that the class of strongly immovable cycles of a fixed dimension forms a group so our result applies to new examples.