We prove that if (M n , g), n ≥ 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold:This is implied by the following result:Let (M 2n , g) be a compact, locally irreducible Kähler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CP n or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative.The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow.

For a compact manifold M of dim M = n ≥ 4, we study two conformal invariants of a conformal class C on M . These are the Yamabe constant Y C (M ) and the L n 2norm W C (M ) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y C (M ) is arbitrarily close to the Yamabe invariant Y (M ), and, at the same time, the constant W C (M ) is arbitrarily large. We study the image of the map YW :We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.Akutagawa, Botvinnik, Kobayashi, Seshadri, The Weyl functional near the Yamabe invariant

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Let 1 and 2 be strongly pseudoconvex domains in C n and f : 1 → 2 an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C 1 map to¯ 1 . We then prove that f | ∂ 1 : ∂ 1 → ∂ 2 is a CR or anti-CR diffeomorphism. It follows that 1 and 2 must be biholomorphic or anti-biholomorphic.

We show that if M = X × Y is the product of two complex manifolds (of positive dimensions), then M does not admit any complete Kähler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.

Let X be an arbitrary complex surface and D a domain in X that has a non-compact group of holomorphic automorphisms. A characterization of those domains D that admit a smooth, weakly pseudoconvex, finite type boundary orbit accumulation point is obtained.

International audienc

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