Properly embedded least area planes in gromov hyperbolic 3-spaces

Abstract

Let X be a Gromov hyperbolic 3-space with cocompact metric, and S 2∞ (X) the sphere at infinity of X. We show that for any simple closed curve ⌈ in S2∞ (X), there exists a properly embedded least area plane in X spanning r. This gives a positive answer to Gabai's conjecture from 1997. Soma has already proven this conjecture in 2004. Our technique here is simpler and more general, and it can be applied to many similar settings. © 2007 American Mathematical Society.