Generalized isothermic lattices

We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Moebius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) cross-ratio of the circular lattice. We derive the analogous condition for our generalized isthermic lattices using Steiner's projective structure of conics and we present basic geometric constructions which encode integrability of the lattice. In particular, we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.Comment: 19 pages, 11 figures; v2. some typos corrected; v3. new references added, higlighted similarities and differences with recent papers on the subjec

Surface markers: An identity card of endothelial cells

All endothelial cells have the common characteristic that they line the vessels of the blood circulatory system. However, endothelial cells display a large degree of heterogeneity in the function of their location in the vascular tree. In this article, we have summarized the expression patterns of a number of well-accepted endothelial surface markers present in normal microvascular endothelial cells, arterial and venous endothelial cells, lymphatic endothelial cells, tumor endothelial cells, and endothelial precursor cells

Holomorphic differentials and Laguerre deformation of surfaces

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T -transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their L-Gauss maps, the T -transforms of L-minimal isothermic surfaces have constant mean curvature H = r in some translate of hyperbolic 3-space H3(−r 2) ⊂ R41, de Sitter 3-space S31(r 2) ⊂ R41, or have mean curvature H = 0 in some translate of a time-oriented lightcone in R41. As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T -transformation of L-isothermic surfaces withholomorphic quartic differential