# Can minimal surfaces be characterized by some universal property?

As objects which are minimal (in some respect), this seems entirely plausible, but I'm not sure what category we should be working in, and what restrictions we would need, to actually have a situation where minimal surfaces would be characterized by a universal property, if they ever can be. An uneducated guess on one possible setup where minimal surfaces would be universal: the objects are surfaces whose boundary is a given simple closed curve, and the morphisms are the area-decreasing isometries - it seems like a minimal surface should be a final object, though we would probably need to introduce an equivalence relation on the morphisms to get the maps to be unique?

I'm also curious about the same question, but for geodesics. Perhaps for them, we would use the collection of paths from point $x$ to point $y$ on a given surface, and use the length-decreasing homotopies?

Being a final object isn't the only option - maybe, for any surface, some kind of map will factor through a minimal surface associated to it?

EDIT: I'm worried this is perhaps too soft a question for MathOverflow - I'm not sure there's really a "right" answer.

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What do you mean by a "surface" in an arbitrary category? –  Harry Gindi Jan 28 '10 at 0:16
But minimal surfaces are also usually only locally area minimizing, too. So I don't see the difference with geodesics. So maybe you should first consider length minimizing geodesics? Also, could you provide some motivation about why this characterization would be useful? –  Deane Yang Jan 28 '10 at 0:16
I was trying to come up with an "artificial" category, whose objects are some collection of surfaces in $\mathbb{R}^3$, with some appropriate choice of what the morphisms are, where some universal property characterized being a minimal surface. And my motivation was just curiosity - (as far as I know) nothing would really come of defining them this way. –  Zev Chonoles Jan 28 '10 at 0:23
There's a notion in semi-locally-simply-connected spaces of a "universal cover", but this is the largest cover. It has the property that all covering space projections factor through. This seems like a sort of dual property. –  Harry Gindi Jan 28 '10 at 0:31
Hmmm... that's an interesting property, but I was mainly trying to capture things like geodesics, or lines of curvature, or other differential geometry concepts where some quantity is minimized (or vanishes) in the language of category theory. My roommate, a differential geometry fan, is skeptical that category theory would really help him - which I suppose I agree with, but I would like to see at least how to describe the basic objects of study in a categorical fashion. –  Zev Chonoles Jan 28 '10 at 1:24

I'm not sure if this answer provides you with the universal property that you desire, but there is such a category that unifies these concepts that you are after.

Cohen, Jones and Segal introduced a concept known as the "Flow Category" in the paper Morse Theory and Classifying Spaces, which associates to any manifold with a Morse Function a category whose objects are the critical points of the Morse function and whose morphisms are the gradient trajectories of some gradient-like vector field. Here is the reference:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.5003

You can get the paper on Ralph Cohen's page if you don't have university access:

http://math.stanford.edu/~ralph/papers.html

Recall that Morse Theory was invented by Marston Morse to study geodesics on manifolds. Geodesics correspond precisely to critical points of the Energy functional. I imagine that any variational problem fits into this framework.

As a word of caution, understanding the space of gradient trajectories lies at the heart of Floer Theory, so if you want to understand Morse Theory on infinite dimensional spaces, prepared to get your hands dirty with some serious analysis. Comment if you want more references. Also, most of the above article is concerned with proving a very elegant result about the classifying space of this category for certain situations. It is very slick!

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I was pretty skeptical, but this seems like a promising answer to the question. –  Deane Yang Jan 28 '10 at 2:08
I was skeptical of my idea too, but I agree - this sounds like what I was thinking of! I'm afraid I don't know nearly enough about the geometry aspects of this paper to really get a handle on it, but I can absolutely appreciate the cleverness in making the objects of the category the critical points of a function measuring some property of the manifold. Thanks for a great answer! –  Zev Chonoles Jan 28 '10 at 2:44