Let M be a compact Riemannian manifold, π : \tilde{M} → M be the universal covering and ω be a smooth 2-form on M with π^∗ω cohomologous to zero. Suppose the fundamental group π_1(M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form η on \tilde{M} of linear (resp. bounded) growth such that π^∗ω = dη. As applications, we prove that on a compact Kahler manifold (M,ω) with π^∗ω cohomologous to zero, if π_1(M) is CAT(0) or automatic (resp. hyperbolic), then M is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic (−1)^{dim_RM/2χ(M) ≥ 0 (resp. >0).

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We present an alternate description of the Ozsv´ath-Szab´o contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M, ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.

Let M and N be two compact complex manifolds. We show that if the tautological line bundle O_{T^∗}M(1) is not pseudo-effective and O_{T^∗}N(1) is nef, then there is no non-constant holomorphic map from M to N. In particular, we prove that any holomorphic map from a compact complex manifold M with RC-positive tangent bundle to a compact complex manifold N with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Given an SO(3)-bundle with connection, the associated twosphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c1 = 0 which are never K¨ahler; e.g., we produce such manifolds with b1 = 0 = b3 and also with c2 · [ω] < 0, answering questions posed by Smith–Thomas–Yau [37]. Examples come from Riemannian geometry, via the Levi–Civita connection on +. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells–Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen–Tian [6]. One metric satisfying the curvature inequality is hyperbolic four-space H4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw −yz = 0 in C4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H4 acting symplectomorphically on the resolution and isometries of H3 acting biholomorphically on the smoothing.