Alexandrov's soap bubble theorem states that a connected, compact, smooth embedded surface without boundary in three-dimensional Euclidean space is a round sphere if its mean curvature is constant (Alexandrov 1962, Meeks et al. 2016). There is no assumption on the genus of the surface.
The proof uses Alexandrov reflection, also called the method of moving planes. A plane is moved inward from outside the surface, and the portion cut off is reflected across the plane. At the first tangential contact, the reflected piece must coincide with the original surface, giving a plane of symmetry. Repeating this argument in every direction characterizes the round sphere (Meeks et al. 2016).
The embedded surface hypothesis is essential. The Wente torus is a compact immersed constant mean curvature surface with self-intersections, and is not a sphere (Meeks et al. 2016).
