Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture

TL;DR

This paper reformulates Caratheodory's conjecture using the properties of graphs of functions with decaying gradients, deriving integral equations for their normal curvatures, and explores the existence of principal lines in these graphs.

Contribution

It introduces a new reformulation of Caratheodory's conjecture in terms of asymptotically flat graphs and derives integral equations related to their normal curvatures.

Findings

Established weaker forms of Caratheodory's conjecture

Proved existence of uncountably many principal lines in certain graphs

Connected geometric conjecture to integral equations via divergence theorem

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.