Existence of minimizing Willmore Klein bottles in Euclidean four-space

Let $K$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles in Euclidean $n$-space is attained by a smooth embedded Klein bottle, where $n\geq 4$. There are three distinct regular homotopy classes of immersed Klein bottles in the Euclidean four-space each one containing an embedding. One is characterized by the property that it contains the minimizer just mentioned. For the other two regular homotopy classes we show that the Willmore energy is bounded from below by $8\pi$. We give a classification of the minimizers of these two classes. In particular, we prove the existence of infinitely many distinct embedded Klein bottles in Euclidean four-space that have Euler normal number $-4$ or $+4$ and Willmore energy $8\pi$. The surfaces are distinct even when we allow conformal transformations of the ambient space. As they are all minimizers in their homotopy class they are Willmore surfaces.Comment: final version, to appear in Geometry & Topolog

On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules

This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible \Lambda-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (\phi,\Gamma)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend it to the (slightly non-commutative) semi-global setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves E over the extension of Q_p which trivialises the p-power division points E(p) of E. In this sense the paper is complimentary to the joint work [7] on noncommutative Main Conjectures for CM elliptic curves.Comment: 39 pages, revised version, previous title was "On the non-commutative Local Main Conjecture for elliptic curves with complex multiplication

Field dependent anisotropy change in a supramolecular Mn(II)-[3x3] grid

The magnetic anisotropy of a novel Mn(II)-[3x3] grid complex was investigated by means of high-field torque magnetometry. Torque vs. field curves at low temperatures demonstrate a ground state with S > 0 and exhibit a torque step due to a field induced level-crossing at B* \approx 7.5 T, accompanied by an abrupt change of magnetic anisotropy from easy-axis to hard-axis type. These observations are discussed in terms of a spin Hamiltonian formalism.Comment: 4 pages, 4 figures, to be published in Phys. Rev. Let