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The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature

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Abstract

We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki–Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension \(2n+1\) and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds \(M_{l_1,l_2,\mathbf{w}}\) of dimension \(2n+3\) which depend on four integer parameters \(l_1,l_2,w_1,w_2\). Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two-dimensional subcone of Sasaki–Ricci solitons in the Sasaki cone, and a unique Sasaki–Einstein metric in each of the two-dimensional subcones.

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Notes

  1. In particular, there we chose \(w_1\le w_2\); whereas, here we use the opposite convention, \(w_1\ge w_2\).

  2. See also this proof for the scaling factor between the admissible metric and the resulting transverse metric. The reciprocal of this scaling factor applies to the lift of \(V\) above and thus it is easy to see that the resulting vector field (which is basically just a multiple of \(H_1\)) on the Sasaki manifold depends smoothly on \(v_1\) and \(v_2\) as well. (see also Lemma 7.1 in [12]).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Charles P. Boyer

  2. Department of Mathematics, Union College, Schenectady, NY, 12308, USA

    Christina W. Tønnesen-Friedman

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  1. Charles P. Boyer
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  2. Christina W. Tønnesen-Friedman
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Correspondence to Charles P. Boyer.

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Both authors were partially supported by grants from the Simons Foundation, CPB by (#245002) and CWT-F by (#208799).

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Boyer, C.P., Tønnesen-Friedman, C.W. The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature. J Geom Anal 26, 1023–1060 (2016). https://doi.org/10.1007/s12220-015-9583-9

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