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Cauchy problems for discrete affine minimal surfaces

Marcos Craizer, Thomas Lewiner, Ralph Teixeira (2012)

Archivum Mathematicum

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...

Centroaffine differential geometry and its relations to horizontal submanifolds

Luc Vrancken (2002)

Banach Center Publications

We relate centroaffine immersions f : M ⁿ → ℝ n + 1 to horizontal immersions g of Mⁿ into S n + 1 2 n + 1 ( 1 ) or H n 2 n + 1 ( - 1 ) . We also show that f is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if g is minimal.

Characterization of compact subsets of curves with ω-continuous derivatives

Marcin Pilipczuk (2011)

Fundamenta Mathematicae

We give a characterization of compact subsets of finite unions of disjoint finite-length curves in ℝⁿ with ω-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set K iff every triple of points of K behaves like a triple of points of a regular curve. This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of...

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