New Einstein Metrics in Dimension Five

By Charles P. Boyer, Krzysztof Galicki. 2000

82 citations recorded by the source corpus.

The purpose of this note is to introduce a new method for proving the existence of Sasakian-Einstein metrics on certain simply connected odd dimensional manifolds. We then apply this method to prove the existence of new Sasakian-Einstein metrics on $\scriptstyle{S^2\times S^3}$ and on $\scriptstyle{(S^2\times S^3)# (S^2\times S^3).}$ These give the first known examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Koll\'ar, AG/9910118, who obtained new examples of K\"ahler-Einstein del Pezzo surfaces with quotient singularities.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4364a7f8e416559e1c4a45b9a8134a79e55ace38

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Toric Sasaki-Einstein metrics on S2 × S3 (2004, background): “Recently [5, 8, 9, 6], a rich set of examples of quasi-regular Sasaki–Einstein metrics have been shown to exist on #l(S(2) ×S3) with l = 1, .”
Einstein metrics on five-dimensional Seifert bundles (2005, background): “There are only a handful of cases when L --+ X is a circle bundle and the above theorem applies, but Boyer and Galicki [1, 2 ] observed that many more cases appear when X is allowed to be an orbifold.”
Sasaki-Einstein Manifolds (2010, methodology): “This was first studied in the case of Fano orbifolds by Demailly-Kollár [38], and indeed it is their results that led to the first examples of quasiregular Sasaki-Einstein 5-manifolds in [17].”
Sasaki–Einstein metrics and K–stability (2015, methodology): “Estimates for the α-invariant were also used by Boyer-Galicki [12, 13], Boyer-Nakamaye [17], Kollár-Johnson [54], Ghigi-Kollár [47], Kollár [58, 56] and others to produce many infinite families of Sasaki-Einstein metrics in dimensions 5 and 7, and higher.”
Examples of asymptotically conical Ricci-flat Kähler manifolds (2008, background): “Two families of hypersurface examples are given which are distinguished by the condition b3(Y ) = 0 or b3(Y ) 6= 0.”
Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals (2004, background): “It was only recently [29, 30, 31, 32] that quasi–regular Sasaki–Einstein metrics were shown to exist on #l(S(2) × S(3)) with l = 1, .”
The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds (2005, background): “and Galicki [ 14 ], quasi–regular Sasaki–Einstein metrics have been shown to exist on #l(S2 × S3) with l = 1, . . . ,9. The irregular case is perhaps more interesting since so”