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Given an SO(3)-bundle with connection, the associated two-sphere bundlecarries a natural closed 2-form. Asking that this be symplectic gives acurvature inequality first considered by Reznikov. We study this inequality inthe case when the base has dimension four, with three main aims. Firstly, weuse this approach to construct symplectic six-manifolds with c_1=0 which arenever Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also withc_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come fromRiemannian geometry, via the Levi-Civita connection on Lambda^+. The underlyingsix-manifold is then the twistor space and often the symplectic structure tamesthe Eells-Salamon twistor almost complex structure. Our second aim is toexploit this to deduce new results about minimal surfaces: if a certaincurvature inequality holds, it follows that the space of minimal surfaces (withfixed topological invariants) is compactifiable; the minimal surfaces must alsosatisfy an adjunction inequality, unifying and generalising results ofChen--Tian. One metric satisfying the curvature inequality is hyperbolicfour-space H^4. Our final aim is to show that the corresponding symplecticmanifold is symplectomorphic to the small resolution of the conifold xw-yz=0 inC^4. We explain how this fits into a hyperbolic description of the conifoldtransition, with isometries of H^4 acting symplectomorphically on theresolution and isometries of H^3 acting biholomorphically on the smoothing.

Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold