Space of embedded surfaces with a common point Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this space such that it is sequentially compact (and Haussdorf) and if $ M_n $ is convergent to $ M $ then $ H_n \rightarrow H $ and $ |A_n(0)| \rightarrow |A(0)| $ where $ H_n,H $ are the mean curvatures and $ A_n,A $ are the second fundamental forms.
Thanks
 A: Presumably you want a bound on the mean curvature, otherwise the  sequence of spheres $(x-1/n)^2+y^2+z^2=1/n^2$ will be problematic: it has no convergent subsequence in any reasonable sense. 
Already  the case $H_n\equiv 0$ is very complicated. To begin with, think of a sequence of catenoids with neck size shrinking to zero. If the point that you fixed sits on the neck, you will see both principal curvatures blowing up: one to $+\infty$, the other to $-\infty$. This survey by Minicozzi lists other things that can happen with a sequence of minimal surfaces. 
To suggest a direction in which a positive result may be found: up to rotation, the surface is given near the origin by equation $z=f(x,y)$ where $f(0,0)=0$ and  $f$ is real-analytic. Impose a uniform lower bound on the radius of disk $x^2+y^2<r^2$ in which $f$ is defined. Also impose a uniform upper bound on $|f|$. Then the interior regularity for mean curvature equation (Corollary 16.7 in the book by Glbarg & Trudinger) yields a uniform upper bound on Hessian of $f$ at $0$. This allows you to select a subsequence for which the second fundamental form at $0$ converges. 
