The level set flow of a hypersurface in R4\mathbb R^4 of low entropy does not disconnect

We show that if $\Sigma\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $\lambda(\Sigma)\leq \lambda(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $\Sigma$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.Comment: Strengthened statement of Proposition 3.4 and fixed a gap in proof of Proposition 3.

A Remark on a Uniqueness Property of High Multiplicity Tangent Flows in Dimension Three

In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive integer multiple of a shrinking plane, cylinder or sphere, then, modulo rotations, all tangent flows at the point are the same.Comment: 6 pages, 0 figur

Electronic structure and transport properties of atomic NiO spinvalves

Ab-initio quantum transport calculations show that short NiO chains suspended in Ni nanocontacts present a very strong spin-polarization of the conductance. The generalized gradient approximation we use here predicts a similiar polarization of the conductance as the one previously computed with non-local exchange, confirming the robustness of the result. Their use as nanoscopic spinvalves is proposed.Comment: 2 pages, 1 figure; accepted in JMMM (Proceedings of ICM'06, Kyoto