Caner Koca

Vanderbilt University, Нашвилл, Michigan, United States

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Publications (8)2.49 Total impact

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    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 conformal class.
    Preview · Article · Nov 2015
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    Mustafa Kalafat · Caner Koca
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    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
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    Mustafa Kalafat · Caner Koca
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    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 possibilities.
    Full-text · Article · May 2013 · Houston journal of mathematics
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    Mustafa Kalafat · Caner Koca
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    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.
    Full-text · Article · Jun 2012
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    Caner Koca
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    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 Hirzebruch surfaces.
    Preview · Article · Dec 2011 · Journal of Geometry and Physics
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    Caner Koca
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    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
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    Caner Koca
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    ABSTRACT: ABSTRACT ORBITS IN THE ANTI-INVARIANT SUBLATTICE OF THE K3-LATTICE Caner Koca
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    CANER KOCA
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    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.
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