Are compact constant mean curvature hypersurface sphere?

Question

I am recently reading a paper. The author seems to use that (293 page) for $n\ge 2$, if $M^n\subset \mathbb R^{n+1}$ has constant mean curvature and $M^n$ is compact, then $M^n$ is sphere. But I google the constant mean curvature manifold, there are some constant mean curvature tori. What is my misunderstanding?

Answer 1

From the notation $M^n \subset \mathbb R^{n+1}$ it is assumed that $M^n$ is embedded in $\mathbb R^{n+1}$. Then $M^n$ is the round sphere. This is proved by A. D. Alexandrov. See here.

For the classical dimension, the only CMC surface of genus $0$ in $\mathbb R^3$ is the round sphere. This is proved by H. Hopf. See here. Assumptions on embeddedness is not needed.

Whether or not there exists CMC surface of higher genus was called Hopf conjecture and the first example was found by Wente. See here. The examples of course are not embedded.