Volume 2010, Issue 643

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat spin manifolds. This class consists of manifolds obtained by factoring out the upper half-space of by congruence subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.

Let N be a compact domain with weakly two-convex boundary ∂N in a Riemannian 4-manifold M with nonnegative isotropic curvature. If D is a stable minimal disk in N with ∂D ⊂ ∂N that solves the free boundary problem, then D is infinitesimally holomorphic; moreover, it is ± holomorphic if M is a Kähler surface with positive scalar curvature, and it is holomorphic for some complex structure if M is a hyperkähler surface. We also show that if N is a compact domain in M of dim M ≧ 4 with nonnegative isotropic curvature and ∂N is two-convex, then π 1 ( ∂N ) → π 1 ( N ) is injective.

We consider a closed manifold M with a Riemannian metric g ij ( t ) evolving by ∂ t g ij = –2 S ij where S ij ( t ) is a symmetric two-tensor on ( M , g ( t )). We prove that if S ij satisfies the tensor inequality 𝒟( S ij , X ) ≧ 0 for all vector fields X on M , where 𝒟( S ij , X ) is defined in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow ∂ t g ij = –2 S ij . In the case where S ij = R ij , the Ricci curvature of M , the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow presented in [The entropy formula for the Ricci flow and its geometric applications, 2002]. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system developed in [Evolution of an extended Ricci flow system, AEI Potsdam, 2005], the Ricci flow coupled with harmonic map heat flow presented in [Müller, The Ricci flow coupled with harmonic map heat flow, ETH Zürich, 2009], and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK -theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K -theoretic information about certain graph C *-algebras.

We construct rigid étale coverings of certain p -adic period domains, and show that they are defined by p -divisible groups.

Let G be a totally disconnected, locally compact group admitting a contractive automorphism α . We prove a Jordan-Hölder theorem for series of α -stable closed subgroups of G , classify all possible composition factors and deduce consequences for the structure of G .

We prove that given any compact group G , there exists a minimal action of G on a II 1 factor M such that the bimodule category of the fixed-point II 1 factor M G is naturally equivalent with the representation category of G . In particular, all subfactors of M G with finite Jones index can be described explicitly.