The regularity of harmonic maps into spheres and applications to Bernstein problems

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial

K-theory for ring C*-algebras - the case of number fields with higher roots of unity

We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number fields.Comment: 26 pages, final version, accepted for publication in the Journal of Topology and Analysi

The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension

We identify a region \Bbb{W}_{\f{1}{3}} in a Grassmann manifold \grs{n}{m}, not covered by a usual matrix coordinate chart, with the following important property. For a complete $n-$submanifold in \ir{n+m} \, (n\ge 3, m\ge2) with parallel mean curvature whose image under the Gauss map is contained in a compact subset K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m}, we can construct strongly subharmonic functions and derive a priori estimates for the harmonic Gauss map. While we do not know yet how close our region is to being optimal in this respect, it is substantially larger than what could be achieved previously with other methods. Consequently, this enables us to obtain substantially stronger Bernstein type theorems in higher codimension than previously known.Comment: 36 page

Prevention of the disrupted enamel phenotype in Slc4a4-null mice using explant organ culture maintained in a living host kidney capsule.

Slc4a4-null mice are a model of proximal renal tubular acidosis (pRTA). Slc4a4 encodes the electrogenic sodium base transporter NBCe1 that is involved in transcellular base transport and pH regulation during amelogenesis. Patients with mutations in the SLC4A4 gene and Slc4a4-null mice present with dysplastic enamel, amongst other pathologies. Loss of NBCe1 function leads to local abnormalities in enamel matrix pH regulation. Loss of NBCe1 function also results in systemic acidemic blood pH. Whether local changes in enamel pH and/or a decrease in systemic pH are the cause of the abnormal enamel phenotype is currently unknown. In the present study we addressed this question by explanting fetal wild-type and Slc4a4-null mandibles into healthy host kidney capsules to study enamel formation in the absence of systemic acidemia. Mandibular E11.5 explants from NBCe1-/- mice, maintained in host kidney capsules for 70 days, resulted in teeth with enamel and dentin with morphological and mineralization properties similar to cultured NBCe1+/+ mandibles grown under identical conditions. Ameloblasts express a number of proteins involved in dynamic changes in H+/base transport during amelogenesis. Despite the capacity of ameloblasts to dynamically modulate the local pH of the enamel matrix, at least in the NBCe1-/- mice, the systemic pH also appears to contribute to the enamel phenotype. Extrapolating these data to humans, our findings suggest that in patients with NBCe1 mutations, correction of the systemic metabolic acidosis at a sufficiently early time point may lead to amelioration of enamel abnormalities

The Gauss image of entire graphs of higher codimension and Bernstein type theorems

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson-Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.Comment: 28 page