An integral formula for closed surfaces and a generalization of Hp- theorem. (English) Zbl 0651.53042

Rotondaro, Giovanni

Commentat. Math. Univ. Carol. 29, No. 2, 253-254 (1988).

Let F be a closed, immersed, and orientable \(C^{\infty}\) surface in \({\mathbb{R}}^ 3\). The author shows that if \(Hp=1\) on F, where H is the mean curvature, and p is the support function, then F is a standard sphere. Previously this result has been proved under the additional assumption that the Gauss curvature of F is positive.

Reviewer: V.Oliker

MSC:

53C40	Global submanifolds
53A05	Surfaces in Euclidean and related spaces
53C45	Global surface theory (convex surfaces à la A. D. Aleksandrov)

Keywords:

integral formula; rigidity theorem; mean curvature; support function; standard sphere