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In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is the following: Can a sequence of constant mean curvature (non-zero a prior) converge (in $C^2$- topology) to a branch immersion if we impose a condition on the curvature of the boundary such that $$\int \kappa d\sigma< 4\pi .$$

Indeed, there is some regularity results for minimal surface which exclude branch point under this kind of assumption, see ยง377 of the book of Nitsche on minimal surface, moreover here the mean curvature can be made small by rescaling the ambient space since the condition on the the total curvature is scale-invariant. My feeling is that it is impossible to smoothly converge to a branch immersion without a complicated boundary or curvature.... but i have no proof.

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By '$\kappa$', in your formula, do you mean the geodesic curvature (as measured in the surface) or the extrinsic curvature of the boundary curve as it is embedded in $3$-space? – Robert Bryant Jan 15 '12 at 13:36
i mean the curvature of the curvature of the boundary curve seen as an embedded curve of $\R^3$. In Your example it is something like $2\pi(2k+1)$. – Paul Jan 15 '12 at 13:56

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