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We show that Caratheodory's conjecture, on umbilical points of closed convexsurfaces, may be reformulated in terms of the existence of at least one umbilicin the graphs of functions f: R^2-->R whose gradient decays uniformly fasterthan 1/r. The divergence theorem then yields a pair of integral equations forthe normal curvatures of these graphs, which establish some weaker forms of theconjecture. In particular, we show that there are uncountably many principallines in the graph of f whose projection into R^2 are parallel to any givendirection.

Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture