Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture
Differential Geometry
2011-08-30 v2 Dynamical Systems
Geometric Topology
Abstract
We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of f whose projection into R^2 are parallel to any given direction.
Keywords
Cite
@article{arxiv.1101.3031,
title = {Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture},
author = {Mohammad Ghomi and Ralph Howard},
journal= {arXiv preprint arXiv:1101.3031},
year = {2011}
}
Comments
Final draft. Accepted for publication in Proceedings of the AMS