Canonical Sasakian Metrics

By Charles P. Boyer, Krzysztof Galicki, Santiago R. Simanca. 2006

108 citations recorded by the source corpus.

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b6d9d5c37cfa1f7ce3f8fb696487e96cecb0082a

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In Applied Mathematics

1. J. Sparks
2. Craig van Coevering
3. W. Minicozzi
4. T. Colding
5. Song Sun
6. Misha Verbitsky
7. L. Ornea
8. W. H. Meeks

In Geometry and Topology

1. Craig van Coevering
2. J. Sparks
3. W. Minicozzi
4. T. Colding
5. Song Sun
6. Misha Verbitsky
7. L. Ornea
8. A. Futaki

Citations in the corpus

• Sasakian structures on CR-manifolds (2006, methodology): “Proof: The last claim of Theorem 1.11 is due to [ BGS ]; we give a new proof of this statement in Subsection 6.1, and prove the rest of Theorem 1.11.”
• Sasaki-Einstein Manifolds (2010, background): “This result was extended [48, 28] to obstructions to the existence of Sasakian metrics with harmonic basic kth Chern form, again generalizing the Kähler result to the transverse setting.”
• On uniqueness of tangent cones for Einstein manifolds (2012, background): “ges to zero like (logr)−β for some β > 0, where r is the distance to a fixed point. 1 2 TOBIASHOLCKCOLDINGANDWILLIAMP.MINICOZZIII others, and there is a extensive literature of examples; see, e.g., [BGS], [DS], [K1], [K2], [MS1], [MS2], [MSY1], [MSY2], [TY1] and [TY2]. Most examples fall into seve”
• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds (2006, background): “l study in later sections the deformations of transverse Kahler structures. They correspond to the deformations of η fixing ξ. A fixed choice of ξ is called the 4 polarization of the Sasaki manifold in [9]. As ηvaries the contact bundle Dvaries, while we fix the transverse holomorphic structure of the ”
• Examples of asymptotically conical Ricci-flat Kähler manifolds (2008, background): “t, the closure of this subgroup is a torus Tk ⊂ Aut(S) where k= rank(S). Choose a vector field ζin the integral lattice of the Lie algebra t of Tk, ζ∈ ZT ⊂ t, and such that η(ζ) >0 on S. Then as in [13] on can show there is a unique Sasaki structure (˜g,ζ,η,˜ Φ) with the same˜ CR-structure and Ree”