It is a well known result of Alexandrov that the *only compact, connected, constant mean curvature surface is the ball*.
There is a generalized notion of curvature known as

*generalized mean curvature*which makes sense on rectifiable varifolds. In particular if $V$ is a recifiable varifold than the generalized mean curvature is defined as follows:

Let $\|\delta V\|$ denote the first variation of $V$. Then if this measure is absolutely continuous with respect to $\|V\|$ then by the Riesz representation theorem:

$ \delta V(g) = - \int g \cdot H d\|V\|$, for $H$ defined $\|V\|$ a.e.

- H as defined here is the
*generalized mean curvature*.

**Question:** Working in the plane, $\mathbb{R}^2$, assume that $\Omega$ is a set of bounded variation so $|\partial \Omega| < C < +\infty$ and define $H$ as above.

If $H=$ constant, what does this imply about $\Omega$? Must $\Omega$ be a ball? Are there examples of singular $BV$ sets which satisfy this condition but are not balls?

What is the generalized mean curvature of a square?