Limit lamination theorem for H-disks

We describe the lamination limits of sequences of compact disks $$M_n$$ M n embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature $$H_n$$ H n , when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi (Ann Math 160:573–615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi (Ann Math 167:211–243, 2008) for minimal disks.

Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface

Let L be an oriented Lagrangian submanifold in an n-dimensional Kähler manifold M. Let u: D → M be a minimal immersion from a disk D with u(𝜕D) ⊂ L such that u(D) meets L orthogonally along u(𝜕D). Then the real dimension of the space of admissible holomorphic variations is at least n + μ(E, F), where μ(E, F) is a boundary Maslov index; the minimal disk is holomorphic if there exist n admissible holomorphic variations that are linearly independent over ℝ at some point p ∈ 𝜕D; if M = ℂPn and u intersects L positively, then u is holomorphic if it is stable, and its Morse index is at least n + μ(E, F) if u is unstable.