An affine invariant notion of minimal surface?

Question

The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $T(M)$ is not a minimal surface for most $T\in\mathrm{GL}(\Bbb R^n)$. This is because having zero mean curvature is not invariant under general linear transformations.

But is there perhaps a different notion of minimal surface that is affinely invariant? Ideally this other notion would share many of the nice properties with the usual notion or would have a similar definition, e.g. minimizing some integral of a locally defined property (such as Dirichlet energy for usual minimal surfaces).

My motivation: I am given an embedded $m$-sphere $S\subset\Bbb R^n,n>m$ and I need to find a canonical $(m+1)$-dimensional hypersurface with this sphere as boundary. I would have hoped to find a notion of such a surface that transformes well under linear maps.

Answer 1

Perhaps you are looking for the notion of an affine sphere? They are characterized by the affine shape operator being a constant multiple of the identity, are affine differential-geometric objects, and hyperbolic affine spheres are flexible enough to make use of in various circumstances (the foundational results on this are due to Cheng and Yau).

In particular, to every bounded convex domain $\Omega$ in $\mathbb{RP}^n$, there is a unique (up to the action of the affine group) hyperbolic affine sphere in $\mathbb{R}^{n+1}$ with constant affine shape operator $-\text{Id}$ that is asymptotic to $\partial \Omega$.

A reference I am fond of for affine differential geometry is Nomizu's book, and Loftin has a notable survey on affine spheres.

Answer 2

This is an interesting question. I don't know of any way to define an affine invariant surface area for an arbitrary surface. However, if the surface is a closed convex surface, there are several different global affine invariants that could be called an affine invariant analogue of surface area and that satisfy an affine invariant isoperimetric inequality, generalizing the classical Euclidean one. Whereas equality holds for the Euclidean inequality if and only if the surface is a standard sphere, equality holds for the affine inequality if and only if the surface is the boundary of an ellipsoid.