A complex Euclidean reflection group with a non-positively curved complement complex

2021 ◽  
Author(s):  
Ben Coté ◽  
Jon McCammond
Keyword(s):  
Reflection Group ◽  
Positively Curved

scholarly journals Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

2015 ◽  
Vol 37 (3) ◽  
pp. 939-970 ◽  
Author(s):  
RUSSELL RICKS
Keyword(s):  
Geodesic Flow ◽  
Isometric Action ◽  
Structural Result ◽  
Curved Manifolds ◽  
Rank One ◽  
Unit Speed ◽  
Flat Strip ◽  
Geodesically Complete ◽  
Positively Curved ◽  
General Technical

Let$X$be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group$\unicode[STIX]{x1D6E4}$with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of$X$modulo the$\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in$\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when$X$has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when$X$is a tree with all edge lengths in$c\mathbb{Z}$for some$c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

scholarly journals Almost-rigidity and the extinction time of positively curved Ricci flows

2016 ◽  
Vol 369 (1-2) ◽  
pp. 899-911 ◽  
Author(s):  
Richard H. Bamler ◽  
Davi Maximo
Keyword(s):  
Extinction Time ◽  
Ricci Flows ◽  
Positively Curved ◽  
Almost Rigidity

scholarly journals Isometry groups of non-positively curved spaces: structure theory

2009 ◽  
Vol 2 (4) ◽  
pp. 661-700 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Nicolas Monod
Keyword(s):  
Structure Theory ◽  
Isometry Groups ◽  
Curved Spaces ◽  
Positively Curved

scholarly journals The singular terms in the fundamental matrix of crystal optics

2011 ◽  
Vol 467 (2133) ◽  
pp. 2663-2689 ◽  
Author(s):  
Peter Wagner
Keyword(s):  
Explicit Formula ◽  
Fundamental Matrix ◽  
Singular Part ◽  
Wave Surface ◽  
Crystal Optics ◽  
Cauchy Principal ◽  
Singular Terms ◽  
Negatively Curved ◽  
Principal Value ◽  
Positively Curved

We derive an explicit formula for the singular part of the fundamental matrix of crystal optics. It consists of a singularity remaining fixed at the origin x =0, of delta terms located on the positively curved parts of the wave surface, the well-known Fresnel surface and of a Cauchy principal value distribution on the negatively curved part of the wave surface.

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