# An affine invariant notion of minimal surface?

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.

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 $$\partial \Omega$$.
• It is not clear from my phrasing, but I am actually considering $m$-spheres $S$ embedded in $\Bbb R^n$ where $n$ can be much larger than $m$. Does it help if $S\subset\Bbb S^{n-1}\subset\Bbb R^n$? May 24 at 23:13