On the moduli space of positive Ricci curvature metrics on homotopy spheres

We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.Comment: 28 pages, 11 figures. The text has been substantially re-written to improve the expositio

Positive Ricci curvature on highly connected manifolds

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.Comment: Corrected some minor typos and changed document class to amsart. The new document class added 10 pages, so the paper is now now 46 page

On Ptolemaic metric simplicial complexes

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.Comment: 13 page

A generalization of the Perelman gluing theorem and applications

We prove a gluing result that allows to glue two Riemannian manifolds of positive intermediate Ricci curvature along their boundaries, provided the boundaries are isometric and the sum of second fundamental forms is positive semi-definite. This holds in particular for positive sectional curvature and generalizes a result of Perelman for positive Ricci curvature. As application we derive a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary, and obtain results on the observer moduli space of metrics of positive intermediate Ricci curvature on the sphere.Comment: 21 page

On Ptolemaic metric simplicial complexes

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy’s inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions

Slip statistics of dislocation avalanches under different loading modes

Slowly compressed microcrystals deform via intermittent slip events, observed as displacement jumps or stress drops. Experiments often use one of two loading modes: an increasing applied stress (stress driven, soft), or a constant strain rate (strain driven, hard). In this work we experimentally test the influence of the deformation loading conditions on the scaling behavior of slip events. It is found that these common deformation modes strongly affect time series properties, but not the scaling behavior of the slip statistics when analyzed with a mean-field model. With increasing plastic strain, the slip events are found to be smaller and more frequent when strain driven, and the slip-size distributions obtained for both drives collapse onto the same scaling function with the same exponents. The experimental results agree with the predictions of the used mean-field model, linking the slip behavior under different loading modes