| Abstract: |
| Let $M$ be the Euclidean space or the hyperbolic space.
A celebrated theorem of Alexandrov asserts that spheres are the only closed constant mean curvature hypersurfaces embedded in $M$. The talk mainly focuses on the following quantitive version of the Alexandrov theorem:
\medskip
\noindent {\bf Theorem} [Ciraolo - V.]{\bf .} {\em
Let $S$ be an $n$-dimensional, $C^2$-regular, connected, closed hypersurface embedded in $M$. There exist constants $ \epsilon,\, C>0$ such that if
$$
{\rm osc}(H) \leq \epsilon,
$$
then there are two concentric balls $B_{r_i}$ and $B_{r_e}$ such that
$$
S \subset \overline{B}_{r_e} \setminus B_{r_i},
$$
and
$$
r_e-r_i \leq C{\rm osc}(H).
$$
The constants $ \epsilon$ and $C$ depend only on $n$ and upper bounds on the $C^2$-regularity and the area of $S$. }
\medskip
The proof of the theorem makes use of a quantitive study of the method of the moving planes and the result implies a new pinching theorem for hypersurfaces. Furthermore, the theorem is optimal in a sense that it will be specified in the talk.
The last part of the talk will be about an on-going study on the generalization of the result. |
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