We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these y…
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No minimizer exists for certain spherical metrics with edge-cones.
problem Existence of Yamabe minimizers on singular spheres.
method Analyzing standard edge-cone spherical metrics of cone angles greater than or equal to .
result No minimizer exists for the specified spherical metrics.
Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dim…
In this note, we propose a new approach to solving the Calabi problem on manifolds with edge-cone singularities of prescribed angles along complex hypersurfaces. It is shown how the classical approach of Aubin-Yau in derving {\it a priori} estimates for the complex hessian can be made to work via adopting a \emph{good …
We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…
We prove that a certain class of ALE spaces always has a Kahler conformal compactification, and moreover provide explicit formulas for the conformal factor and the Kahler potential of said compactification. We then apply this to give a new and simple construction of the canonical Bochner-Kähler metric on certain weight…
The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twis…
It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…