On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$
V. G. Dmitriev
Abstract:
It is proved that a hypersurface
$F$ imbedded in
$\mathbf R^{n+1}$,
$n\geqslant2$, which is locally convex at all points except for a closed set
$E$ with
$(n-1)$-dimensional Hausdorff measure
$\mathscr H_{n-1}(E)$, and strictly convex near
$E$ is in fact locally convex everywhere. The author also gives various corollaries. In particular, let
$M$ be a complete two-dimensional Riemannian manifold of nonnegative curvature
$K$ and
$E\subset M$ a closed subset for which
$\mathscr H_1(E)=0$. Assume further that there exists a neighborhood
$U\supset E$ such that
$K(x)>0$ for
$x\in U\setminus E$,
$f\colon M\to\mathbf R^3$ is such that
$f|_{U\setminus E}$ is an imbedding, and
$f|_{M\setminus E}\in C^{1,\alpha}$,
$\alpha>2/3$. Then
$f(M)$ is a complete convex surface in
$\mathbf R^3$. This result is an generalization of results in the paper reviewed in RZh Mat, 1973, 7A724.
Bibliography: 19 titles.
UDC:
514.873
MSC: Primary
53C42; Secondary
52A20 Received: 19.02.1980 and 25.11.1980