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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; however. Perhaps for them, it seems less likely that geodesics satisfy we would use the collection of paths from point $x$ to point $y$ on a universal propertygiven surface, since they are and use the length-decreasing homotopies?

Being a final object isn't the only locally length-minimizing.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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Can minimal surfaces be defined characterized by some universal propertiesproperty?

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 make this truehave a situation where minimal surfaces would be characterized by a universal property, if it is true at allthey 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 - would it seems like a minimal surface should be a final object? Would , though we would probably need to introduce an equivalence relation on the morphisms to get the maps to be unique?(Note that Plateau's problem guarantees the existence of at least one such minimal surface)

I'm also curious about the same question, but for geodesics; however, it seems less likely that geodesics satisfy a universal property, since they are only locally length-minimizing.

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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Can minimal surfaces be defined by universal properties?

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 make this true, if it is true at all. 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 - would a minimal surface be a final object? Would we need to introduce an equivalence relation on the morphisms to get the maps to be unique? (Note that Plateau's problem guarantees the existence of at least one such minimal surface)

I'm also curious about the same question, but for geodesics; however, it seems less likely that geodesics satisfy a universal property, since they are only locally length-minimizing.