Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture

Details Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture

2011-01-15

arxiv.org/abs/1101.3031v2

Mathematics

Differential Geometry

Final draft. Accepted for publication in Proceedings of the AMS

Scientific paper

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.

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