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I'm curious about a general answer for oblique planes, but specifically, I'm interested in the case where one circle's axis is perpendicular to the other's, and its center lies on the other's axis. To be precise, let $C_1$ be the unit circle in the $XY$ plane, and $C_2$ be a circle of radius $r$, center $(0, 0, h)$, with axis parallel to the $x$ axis.

Thinking of these two circles as a sort of minimalist sketch of a signet ring, what is the minimal surface that might be thought of as the convex hull of the ring?

I'm hoping for an analytical solution, but also curious about answering this kind of question computationally.

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    $\begingroup$ Do you define the surface via some special parametrizations of two circles? Then, could you specify these parametrizations, or else, how do you define your surface? $\endgroup$ – Wlod AA Dec 19 '18 at 22:03
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    $\begingroup$ @monguin: I think Wlod AA is talking about the parametrization of the circles, not of the minimal surface joining them. $\endgroup$ – Qfwfq Dec 19 '18 at 23:09
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    $\begingroup$ For doing computations you might want to look into Ken Brakke's Surface Evolver facstaff.susqu.edu/brakke/evolver/evolver.html $\endgroup$ – j.c. Dec 20 '18 at 4:13
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    $\begingroup$ You need to DEFINE your surface (or a class of it). Then the answers will provide CONSTRUCTIONS. $\endgroup$ – Wlod AA Dec 20 '18 at 6:56
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    $\begingroup$ Perhaps, what you mean is the following: given two arbitrary circles separated by a plane, find an arbitrary connected surface which contains these two circles, and has the minimal area among all such surfaces. In particular, what are the conditions for such a minimal surface to exist? $\endgroup$ – Wlod AA Dec 20 '18 at 7:02
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This is a quite subtle question and probably doesn't have an answer without further assumptions (and even then is possibly hard to say much).

First of all, when the circles are too far apart there are no connected minimal surfaces spanning them, just the two flat disk solutions.

When the circles are coaxial and lying in parallel planes, then (if they are close enough) there are annular solutions given by pieces of the catenoid (with axis the same as of the two circles). Note in general one gets two such solutions a stable and an unstable solutions.

When the circles are in parallel planes but not coaxial then there are annular solutions coming from the Riemann family of minimal surfaces (which are all foliated by circles).

Here's where it gets a little subtle: When the circles are coaxial and in parallel planes, then it follows from the moving planes method employed by Schoen ("Uniqueness, symmetry, and embeddedness of minimal surfaces") that the catenoid pieces are the only possible connected minimal surfaces spanning the circles. When the circles are not coaxial, then a classic work of Shiffman ("On Surfaces of Stationary Area Bounded by Two Circles, or Convex Curves, in Parallel Planes") implies that any connected annular surface spanning to two circles is foliated by circles and hence a piece of a Riemann example (a result possible due to Riemann himself). However, it is a open problem whether in this case there are solutions of other topological type (this is a special case of the $4\pi$ Conjecture -- but is still open as far as I know in this simple case). In other words, we don't know enough to say much about even a much simpler version of your problem once we leave the world of minimal annuli.

That being said, I believe Shiffman's result still holds in some form for circles not in parallel planes so it is possible that you can get a fair amount of information in your case for minimal annuli (though whether that is enough for an explicit parameterization is hard to say -- I would guess not).

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