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Reeb vector field
About: Reeb vector field is a research topic. Over the lifetime, 254 publications have been published within this topic receiving 4118 citations.
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TL;DR: The existence of a "Plastikstufe" for a contact structure implies the Weinstein con- jecture for all supporting contact forms as mentioned in this paper, which is the case for all contact structures.
Abstract: The existence of a "Plastikstufe" for a contact structure implies the Weinstein con- jecture for all supporting contact forms.
46 citations
TL;DR: In this article, it was shown that the volume of a Sasaki-Einstein manifold is always an algebraic number, relative to that of the round sphere, and that it is a function on the space of Reeb vector fields.
Abstract: We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.
46 citations
TL;DR: In this article, the authors studied normal CR compact manifolds in dimension 3 and showed that the underlying manifolds are topologically finite quotients of a non-flat circle bundle over a Riemann surface.
Abstract: We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotients of \(S^3\) or of a non-flat circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of \(S^1\), and we classify the normal CR structures on these manifolds.
45 citations
TL;DR: In this paper, the authors classify simply connected compact Sasaki manifolds of dimension 2 n + 1 with positive transverse bisectional curvature and show that the Kahler cone corresponding to such manifolds must be bi-holomorphic to C n+1 \ { 0 }.
44 citations
43 citations