Weyl curvature and the Euler characteristic in dimension four

We give lower bounds, in terms of the Euler characteristic, for the $L^2$-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.Comment: 6 page

Totally geodesic discs in strongly convex domains

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let \Omega_i \subset \C^{n_i}, \ i=1,2, be bounded $C^3$ strongly convex domains. If $\phi: (\Omega_1, d^K_{\Omega_1}) \rightarrow (\Omega_2, d^K_{\Omega_2})$ is an isometry, i.e. d^K_\Omega_{n_2}(f(\zeta),f(\eta)) = d^K_{n_1} (\zeta,\eta) for all $\zeta,\eta \in \Omega_1,$ then $\phi$ is either holomorphic or anti-holomorphic.Comment: 12 page

On the topology of manifolds with positive isotropic curvature

We show that a closed orientable Riemannian $n$-manifold, $n \ge 5$, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of $S^{n-1} \times S^1$.Comment: 5 Pages. To appear in Proc. of AM