In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration with smooth central fiber.

We prove several formulas related to Hodge theory and the KodairaSpencerKuranishi deformation theory of Khler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on CalabiYau manifolds and also a construction of global canonical family of holomorphic ( n , 0 ) -forms on the deformation spaces of CalabiYau manifolds. Similar constructions are also applied to the deformation spaces of compact Khler manifolds.

For any n-dimensional smooth manifold $\Sigma$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $\Sigma$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_\infty$ (maybe empty) at infinity. Previously this was shown (i) for n$\leq$7, and (ii) for arbitrary n up to the first singular time without the smooth condition on $M_\infty$.

Motivated by new explicit positive Ricci curvature metrics on the four-sphere which are also Einstein-Weyl, we show that the dimension of the Einstein-Weyl moduli near certain Einstein metrics is bounded by the rank of the isometry group and that any Weyl manifold can be embedded as a hypersurface with prescribed second fundamental form in some Einstein-Weyl space. Closed four-dimensional Einstein-Weyl manifolds are proved to be absolute minima of the L²-norm of the curvature of Weyl manifolds and a local version of the Lafontaine inequality is obtained. The above metrics on the four-sphere are shown to contain minimal hypersurfaces isometric to S¹S² whose second fundamental form has constant length.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.