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@article{2374058,
author = {Gover, A. Rod and Neusser, Katharina and Willse, Travis},
article_number = {April 2024},
doi = {http://dx.doi.org/10.1007/s10231-023-01385-0},
keywords = {Projective differential geometry; Einstein manifolds; Sasaki manifolds; Hyper-Kähler and quaternionic Kähler geometry; Holonomy; Geometric compactifications},
language = {eng},
issn = {0373-3114},
journal = {Annali di Matematica Pura ed Applicata},
title = {Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients},
url = {https://link.springer.com/article/10.1007/s10231-023-01385-0},
volume = {203},
year = {2024}
}
TY - JOUR
ID - 2374058
AU - Gover, A. Rod - Neusser, Katharina - Willse, Travis
PY - 2024
TI - Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients
JF - Annali di Matematica Pura ed Applicata
VL - 203
IS - April 2024
SP - 875-902
EP - 875-902
PB - Springer
SN - 03733114
KW - Projective differential geometry
KW - Einstein manifolds
KW - Sasaki manifolds
KW - Hyper-Kähler and quaternionic Kähler geometry
KW - Holonomy
KW - Geometric compactifications
UR - https://link.springer.com/article/10.1007/s10231-023-01385-0
N2 - We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p, q). Moreover, we show that, if a holonomy reduction to Sp(p, q) of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.
ER -
GOVER, A. Rod, Katharina NEUSSER and Travis WILLSE. Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients. \textit{Annali di Matematica Pura ed Applicata}. Springer, 2024, vol.~203, April 2024, p.~875-902. ISSN~0373-3114. Available from: https://dx.doi.org/10.1007/s10231-023-01385-0.
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