Einstein metrics on spheres

By Charles P. Boyer, Krzysztof Galicki, J. Kollár. 2003

170 citations recorded by the source corpus.

We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e12c05beaccfdc7eca0a66b430a1a49636782e18

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sasaki–Einstein metrics and K–stability (2015, methodology): “the α-invariant [72, 31]. For example, the aﬃne varieties ZBP(p,q) are a special case of the Brieskorn-Pham singularities, which have 3 been thoroughly studied in the literature. Boyer-Galick-Koll´ar [14] used estimates for the α-invariant of Brieskorn-Pham singularities to produce 68 distinct Sasak”
On Eta-Einstein Sasakian Geometry (2004, methodology): “However, as every η-Sasakian metric can be deformed to a Sasakian-Einstein one such obstruction are completely equivalent to the obstructions preventing existence of orbifold K¨ahler-Einstein metric on the transverse space Zf. Existence of K¨ahler-Einstein metrics on various Fano orbifolds can be of”
Landscape of superconducting membranes (2009, background): “According to [42], if the coefficients satisfy the following two conditions, then the link is Sasaki-Einstein: 1 < ∑”
Ricci-flat Kähler metrics on crepant resolutions of Kähler cones (2008, background): “D G ] 2 9 O ct We then consider the case when X = C(S) is toric.”
Canonical Sasakian Metrics (2006, unknown): “D G ] 1 3 M ar”
LOG CANONICAL THRESHOLDS OF SMOOTH FANO THREEFOLDS (2008, background): “Algebraic geometers have been aware of this fact since [21] appeared, and several papers have used it implicitly in recent years (see, for instance, [64] and [65]).”
Sasaki-Einstein Manifolds (2010, methodology): “This was proven in [22], but a simpler method of proof [53] that bypasses the orbifold subtleties is to use Proposition 1.”
Which powers of holomorphic functions are integrable (2008, result): “hic on the closed unit ball B, Z B 1 (ff¯)1−ǫ n dV < ∞ ⇒ Z B 1 (ff¯)1−ǫ dV < ∞ ∀0 < ǫ < ǫn. Various forms of the Gap conjecture are important in the construction of Einstein metrics as in [BGK05]. Ultimately, a gap conjecture type result lies behind the stabilization theorems in [Kol07, ”
Einstein metrics on five-dimensional Seifert bundles (2005, background): “This leads to a large class of new Einstein metrics on various spaces, including spheres and exotic spheres [ 6 ].”
Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “Einstein metrics constructed in [BG00, BGK05 , BGKT05, GK05, BG06] in all odd dimensions also give new Einstein metrics on Cs(N 2n+1 ). For example, one obtains many positive Einstein metrics on � 2n+1 × (0, π) where � 2n+1 is any odd dimen-”