We develop some results from [4] on the positivity of direct image bundles in the particular case of a trivial ¯bration over a one-dimensional base. We also apply the results to study variations of KÄahler metrics.

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensions were proved. We shall explain the results obtained as well as the techniques involved. The potential applications in symplectic topology and mirror symmetry will also be discussed.

We characterize H¨older continuously differentiable m dimensional submanifolds of Euclidean space among m rectifiable sets S in terms of growth conditions on the m density ratios of the Hausdorff measure Hm S.

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if E is an ample vector bundle over a compact Khler manifold E , $ S^ kE\ts\det E $ is both Nakano-positive and dual-Nakano-positive for any E . Moreover, $ H^{n, q}(X, S^ kE\ts\det E)= H^{q, n}(X, S^ kE\ts\det E)= 0$ for any E . In particular, if E is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^ kE\ts\det E, S^ kh\ts\det h) $ is both Nakano-positive and dual-Nakano-positive for any E .