Introduction: Given a plane $(2d)$, two points $(0d)$, and a curve $("1d")$ analogous to a chain (constant length, points can only rotate around their neighbouring points and cant distance or approach themselves to their neighbouring points, and it is non-self intersecting) hanging from the points, under an uniform gravitational field, we have a catenary.
Question: Given the space $(3d)$, two curves $(1d)$, and a surface $(2d)$ (with analogous characteristics, e.g. if the lenght of the chain was fixed before, now the area of the surface is fixed), we have a... what? What is its name?
[Answer: In civil engineering this surface is called "catenary membranes" or "catenary domes"]
Clarification: Illustration of the surface: Imagine I have a pair of curves with finite length, a mesh of inelastic strings (making tiny rectangular cells, for example), and the mesh is hanging from the curves under a uniform gravitational field
A source for physics of the differential equation that leads to the catenary: Exercise 9 on p. 9 of Professor Shiffrin's DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Another source: https://parametricmonkey.com/2015/10/10/catenary-curves/