[Show abstract][Hide abstract] ABSTRACT: This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on
the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann
surfaces of any genus. This generalizes the existence results by C. LeBrun in
arXiv:1411.3992 and arXiv:1504.06669. Moreover, by calculating the (normalized)
Einstein-Hilbert functional of our examples we generalize Theorem E of
arXiv:1504.06669, which speaks to the abundance of Hermitian Einstein-Maxwell
solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly
Hermitian Einstein-Maxwell solutions, first found in arXiv:1504.06669, on the
first Hirzebruch surface in a form which clearly shows that they are conformal
to a common K\"ahler metric. In particular, this yields a non-trivial example
of non-uniqueness of positive constant scalar curvature metrics in a given
[Show abstract][Hide abstract] ABSTRACT: We show that a compact complex surface which admits a conformally K\"ahler
metric g of positive orthogonal holomorphic bisectional curvature is
biholomorphic to the complex projective plane. In addition, if g is a Hermitian
metric which is Einstein, then the biholomorphism can be chosen to be an
isometry via which g becomes a multiple of the Fubini-Study metric.
Full-text · Article · Feb 2015 · Geometriae Dedicata
[Show abstract][Hide abstract] ABSTRACT: We show that if a compact complex surface admits a locally conformally flat
metric, then it cannot contain a 2-sphere of non-zero self intersection. In
particular, the surface has to be minimal. Then we give a list of
Full-text · Article · May 2013 · Houston journal of mathematics
[Show abstract][Hide abstract] ABSTRACT: We show that a compact complex surface together with an Einstein-Hermitian
metric of positive orthogonal bisectional curvature is biholomorphically
isometric to the complex projective plane with its Fubini-Study metric up to
rescaling. This result relaxes the K\"ahler condition in Berger's theorem, and
the positivity condition on sectional curvature in a theorem proved by Koca.
The techniques used in the proof are completely different from theirs.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study a coupled system of equations on oriented compact
4-manifolds which we call the Bach-Merkulov equations. These equations can be
thought of as the conformally invariant version of the classical
Einstein-Maxwell equations in general relativity. Inspired by the work of C.
LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a
variational characterization of solutions to Bach-Merkulov equations as
critical points of the Weyl functional. We also show that extremal Kaehler
metrics are solutions to these equations, although, contrary to the
Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl
functional. We illustrate this phenomenon by studying the Calabi action on
Preview · Article · Dec 2011 · Journal of Geometry and Physics
[Show abstract][Hide abstract] ABSTRACT: In this paper we will prove that the only compact 4-manifold M with an
Einstein metric of positive sectional curvature which is also hermitian with
respect to some complex structure on M, is the complex projective plane CP^2,
with its Fubini-Study metric.
Preview · Article · Dec 2011 · Proceedings of the American Mathematical Society
[Show abstract][Hide abstract] ABSTRACT: This report is aimed to be a user's guide to Morrison's exquisite article, On K3 surfaces with large Picard number, published in Inventiones Mathematicae in 1984.