Let \mathcal M be the space of properly embedded minimal surfaces in ℝ3 with
genus zero and two limit ends, normalized so that every surface M ∊ \mathcal M has
horizontal limit tangent plane at infinity and the vertical component of its flux equals
one. We prove that if a sequence {M(i)}
i
∊ \mathcal M has the horizontal part of the flux bounded from above, then the
Gaussian curvature of the sequence is uniformly bounded. This curvature estimate yields
compactness results and the techniques in its proof lead to a number of consequences,
concerning the geometry of any properly embedded minimal surface in ℝ3 with
finite genus, and the possible limits through a blowing-up process on the scale of
curvature of a sequence of properly embedded minimal surfaces with locally bounded genus
in a homogeneously regular Riemannian 3-manifold.