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You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.

Anyway, see Is there a complete classification of constant mean curvature surfaces? for a beginning.

You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.

Anyway, see Is there a complete classification of constant mean curvature surfaces? for a beginning.

You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.

You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.

Anyway, see Is there a complete classification of constant mean curvature surfaces? for a beginning.

You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces, http://en.wikipedia.org/wiki/Constant-mean-curvature_surface

Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.