On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$

V. G. Dmitriev

Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 451-460

Résumé

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.