192 research outputs found
Hyperk\"ahler ambient metrics associated with twistor CR manifolds
Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR
5-manifolds defined as -bundles over 3-dimensional conformal
manifolds. In this paper, we embed a real analytic twistor CR manifold into the
twistor space of the anti self-dual Poincar\'e-Einstein metric whose conformal
infinity is the base conformal 3-manifold, and construct the associated
Fefferman ambient metric as a neutral hyperk\"ahler metric on the spinor bundle
with the zero section removed. We also describe the structure of the Cheng--Yau
type K\"ahler-Einstein metric which has the twistor CR manifold as the boundary
at infinity.Comment: 31 page
Chains in CR geometry as geodesics of a Kropina metric
With the help of a generalization of the Fermat principle in general
relativity, we show that chains in CR geometry are geodesics of a certain
Kropina metric constructed from the CR structure. We study the projective
equivalence of Kropina metrics and show that if the kernel distributions of the
corresponding 1-forms are non-integrable then two projectively equivalent
metrics are trivially projectively equivalent. As an application, we show that
sufficiently many chains determine the CR structure up to conjugacy,
generalizing and reproving the main result of [J.-H. Cheng, 1988]. The
correspondence between geodesics of the Kropina metric and chains allows us to
use the methods of metric geometry and the calculus of variations to study
chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985]
that locally any two points of a strictly pseudoconvex CR manifolds can be
joined by a chain. Finally, we generalize this result to the global setting by
showing that any two points of a connected compact strictly pseudoconvex CR
manifold which admits a pseudo-Einstein contact form with positive
Tanaka-Webster scalar curvature can be joined by a chain.Comment: are very welcom
強凸領域上のブラシュケ計量の体積繰り込みについて
学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 平地 健吾, 東京大学教授 高山 茂晴, 東京大学教授 小林 俊行, 東京大学教授 古田 幹雄, 東京大学教授 金井 雅彦University of Tokyo(東京大学
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