45 research outputs found
Non trivial examples of coupled equations for K\"ahler metrics and Yang-Mills connections
We provide non trivial examples of solutions to the system of coupled
equations introduced by M. Garc\'ia-Fern\'andez for the uniformization problem
of a triple where is a holomorphic vector bundle over a polarized
complex manifold , generalizing the notions of both constant scalar
curvature K\"ahler metric and Hermitian-Einstein metric.Comment: 17 page
The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics
By combining the join construction from Sasakian geometry with the
Hamiltonian 2-form construction from K\"ahler geometry, we recover
Sasaki-Einstein metrics discovered by physicists. Our geometrical approach
allows us to give an algorithm for computing the topology of these
Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology
rings for several cases of interest and give a formula for homotopy equivalence
in one particular 7-dimensional case. We also show that our construction gives
at least a two dimensional cone of both Sasaki-Ricci solitons and extremal
Sasaki metrics.Comment: 38 pages, paragraph added to introduction and Proposition 4.1 added,
Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular
Sasaki-Einstein structures expanded. Reference adde
The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature
We describe a general procedure for constructing new Sasaki metrics of
constant scalar curvature from old ones. Explicitly, we begin with a regular
Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact
manifold M and construct a sequence, depending on four integer parameters, of
rays of constant scalar curvature (CSC) Sasaki metrics on a compact Sasaki
manifold of dimension . We also give examples which show that the CSC
rays are often not unique on a fixed strictly pseudoconvex CR manifold or a
fixed contact 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 sub cones.Comment: 32 pages. A gap in the argument of applying the admissibility
conditions to irregular Sasakian structures is filled. Some minor corrections
and additions are also made. This is the final version which will appear in
the Journal of Geometric Analysis. It also encorporates much from our paper
arXiv:1309.706