I'm studying a research paper, and there is a theorem whose statement looks very strange to me. In 2001, Gerhard Huisken and Tom Ilmanen proposed a theorem in their collaborative work THE INVERSE MEAN CURVATURE FLOW AND THE RIEMANNIAN PENROSE INEQUALITY:
Main Theorem (Riemannian Penrose Inequality). Let $M$ be a complete, connected $3$-manifold. Suppose that:
(i) $M$ has nonnegative scalar curvature.
(ii) $M$ is asymptotically flat satisfying (0.1) and (0.2), with ADM mass $m$.
(iii) The boundary of $M$ is compact and consists of minimal surfaces, and $M$ contains no other compact minimal surfaces.
Then $$m\geq\sqrt{\frac{|N|}{16\pi}},$$ where $|N|$ is the area of any connected component of $\partial M$. Equality holds if and only if $M$ is isometric to one-half of the spatial Schwarzschild manifold.
Please allow me to omit the conditions (0.1) and (0.2) because they are imposed solely on the Riemannian metric on $M$ and its Ricci curvature.
As indicated by the title, my question lies in the assumption (iii). I know $\partial M$ is a surface in $M$ and therefore qualifies as a candidate for minimal surfaces in $M$, but is it really possible for $\partial M$ to consist of minimal surfaces in $M$? If I take a minimal surface $S$ in $\mathbb{R}^3$ as an example, is it really possible to break $S$ into pieces with each piece being a minimal surface in its own right?
Thank you.
Edit. I know minimal surfaces can be defined in several equivalent ways, but please allow me to adopt the familiar mean-curvature definition, in which a minimal surface is characterized by requiring that its mean curvature vanish at each point. Now, can we join together a family of minimal surfaces smoothly to get another minimal surface? The authors seem to indicate that $\partial M$ is this another minimal surface. Thank you.
