We derive extrinsic curvature estimates for compact disks embedded in R 3 with nonzero constant mean curvature.Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

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We prove a chord arc type bound for disks embedded in R 3 with constant mean curvature that does not depend on the value of the mean curvature. This bound is inspired by and generalizes the weak chord arc bound of Colding and Minicozzi in Proposition 2.1 of [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in R 3 with finite topology or with positive injectivity radius.Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simplyconnected ones. We consider the isometric deformability question for an immersion x : M → R 3 of an oriented non-simply-connected surface with constant mean curvature H. We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of R 3 , either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion x, every cycle in M has vanishing force and, when H = 0, also vanishing torque. Our work generalizes a rigidity result for minimal surfaces [5] to constant mean curvature surfaces (Theorem 5.2). Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque.

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Examples of complete minimal surfaces properly embedded in H 2 × R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H 2 × R, not necessarily proper, which are invariant by a vertical translation or by a hyperbolic or parabolic screw motion. In particular, we construct a large family of non-proper complete minimal disks embedded in H 2 × R invariant by a vertical translation and a hyperbolic screw motion and whose importance is twofold. They have finite total curvature in the quotient of H 2 × R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R 3 satisfying the same properties. They show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H 2 × R.

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