Volume 2013, Issue 685

Abstract. We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold , i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every , there exists a unique (up to left translations) immersed CMC H sphere S H in (Hopf-type theorem). Moreover, this sphere S H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H .

Abstract. We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type , obtained by Serre construction from the triangle singularity . For the Ringel–Schmidmeier classification is thus covered by the classification of vector bundles for tubular type , and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi–Yau categories, indexed by p , which naturally form an ADE-chain.

Abstract. A theorem of Tate asserts that, for an elliptic surface defined over a number field k , and a section , there exists a divisor such that , where is the Néron–Tate height on the fibre above  t . We prove the analogous statement for a one-parameter family of polynomial dynamical systems. Moreover, we compare, at each place of k , the local canonical height with the local contribution to h D , and show that the difference is analytic near the support of D , a result which is analogous to results of Silverman in the elliptic surface context.

Abstract. We prove that the functor of n -fold Pfister forms satisfies purity for regular local rings containing a field of characteristic whose residue field is infinite. As an application we show that purity holds for F 4 -torsors with trivial g 3 invariant.

Abstract. We consider compact minimal surfaces of genus 2 which are homotopic to an embedding. We prove that such surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface and are at most quadratic. For the existence proof of the DPW potential, we give a characterization of stable extensions of spin bundles S by its dual in terms of an associated element of . We also show that the family of holomorphic structures associated to a minimal surface of genus in S 3 is generically stable.

Abstract. For any , we construct properly embedded minimal surfaces in with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.

Abstract. Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the Manin–Mumford and André–Oort Conjecture and is related to a conjecture of Pink. The main technical tool is a new height inequality. We also use it to give another proof of a case of Gubler's result on the Bogomolov Conjecture over function fields.

Abstract. Let k be an algebraically closed field of characteristic p >0. Let be the ring of Witt vectors with coefficients in k . We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the étale cohomology with ℤ p coefficients to the crystalline cohomology with integral coefficients, in the more general context of p -divisible groups endowed with arbitrary families of crystalline tensors over a finite, discrete valuation ring extension of . This extends a result of Faltings. As a main new tool we construct global deformations of p -divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over whose special fibers over k have a Zariski dense set of k -valued points.