Band 2011, Heft 656

We prove the local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point, as proved in [Schnürer, Azouani, Georgi, Hell, Jangle, Köller, Marxen, Ritthaler, Sáez, Schulze and Smith, Trans. Amer. Math. Soc.]. Our theorem is the analog of the result of Grayson [J. Diff. Geom. 26: 285–314, 1987] for curvature flow of closed planar embedded curves.

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkähler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard or a Multi-Taub-NUT manifold. In particular, the underlying complex manifold is either ℂ × ℂ/ℤ or a minimal resolution of a cyclic Kleinian singularity.

We establish a Schubert calculus for Bott–Samelson resolutions in the algebraic cobordism ring of a complete flag variety G / B extending the results of Bressler–Evens [Trans. Amer. Math. Soc. 331: 799–813, 1992] to the algebro-geometric setting.

Bounded local G -shtukas are function field analogs for p -divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport–Zink spaces for p -divisible groups. The underlying schemes of these moduli spaces are affine Deligne–Lusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G -shtuka is isomorphic to the completion of a corresponding affine Deligne–Lusztig variety in that point. This yields bounds on the dimension and proves equidimensionality of the basic affine Deligne–Lusztig varieties.

We investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes. In particular, it is shown that every nilpotent block of a quasisimple group has abelian defect groups. A conjecture of Puig concerning the recognition of nilpotent blocks is also shown for these groups.

We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.

In [Kimura, Math. Ann. 331: 173–201, 2005], we defined the notion of finite dimensional objects in the category of Chow motives. In this paper, we introduce the notion of finite dimensional morphisms for tensor categories so that an object is finite dimensional if and only if its identity morphism is finite dimensional. Applications to Chow motives are discussed.

We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of rational parameters, there are directions in which all trajectories escape, and we prove a rate of escape for almost all directions. These results extend to a dense G δ of parameters.