Geodesic
On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$
Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 451-460 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_1982_42_4_a1,
     author = {V. G. Dmitriev},
     title = {On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$},
     journal = {Sbornik. Mathematics},
     pages = {451--460},
     year = {1982},
     volume = {42},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1982_42_4_a1/}
}
TY  - JOUR
AU  - V. G. Dmitriev
TI  - On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$
JO  - Sbornik. Mathematics
PY  - 1982
SP  - 451
EP  - 460
VL  - 42
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1982_42_4_a1/
LA  - en
ID  - SM_1982_42_4_a1
ER  - 
%0 Journal Article
%A V. G. Dmitriev
%T On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$
%J Sbornik. Mathematics
%D 1982
%P 451-460
%V 42
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1982_42_4_a1/
%G en
%F SM_1982_42_4_a1
V. G. Dmitriev. On the structure of $\mathscr H_{n-1}$-almost everywhere convex hypersurfaces in $\mathbf R^{n+1}$. Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 451-460. http://geodesic.mathdoc.fr/item/SM_1982_42_4_a1/

[1] Kon-Fossen S. E., Nekotorye voprosy differentsialnoi geometrii v tselom, Fizmatgiz, M., 1959 | MR

[2] Pogorelov A. V., Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, M., 1969 | MR

[3] Pogorelov A. V., “K voprosu o regulyarnosti vypukloi poverkhnosti s regulyarnoi metrikoi v evklidovom prostranstve”, DAN SSSR, 139:5 (1961), 1056–1058 | MR | Zbl

[4] Green R. E., Wu H., “On the rigidity of punctured ovaloids”, Ann. Math., 94:1 (1971), 1–20 | DOI | MR

[5] Green R. E., Wu H., “On the rigidity of punctured ovaloids. II”, J. Diff. Geom., 6:4 (1972), 459–472 | MR | Zbl

[6] Gerike H., “Über ein Konvexitatskriterium”, Math. Z., 1937, no. 43, 110–112

[7] Pasqualini L., “Sur les conditions de convexite d'une variété”, Ann. Fac. Sci. Univ. Toulouse, IV s., 2 (1938), 1–45 | MR | Zbl

[8] Pasqualini L., “Sur les conditions de convexite d'une variété close $V_{p-1}p-1$ fois étendue de l'espace euclidien”, Mathematica, Cluj, 16 (1940), 102–108 | MR | Zbl

[9] Burago Yu. D., Zalgaller V. A., “Dostatochnye priznaki vypuklosti”, Zap. nauchn. sem. LOMI, 45 (1974), 3–52 | MR | Zbl

[10] Van Hejenoort J., “On locally convex manifolds”, Comm. Pure Appl. Math., 5 (1952), 223–242 | DOI | MR | Zbl

[11] Shefel S. Z., “$C^1$-gladkie izometricheskie pogruzheniya”, Sib. matem. zh., XV:6 (1974), 1372–1393

[12] Shefel S. Z., “$C^1$-gladkie poverkhnosti ogranichennoi vneshnei polozhitelnoi krivizny”, Sib. matem. zh., XVI:5 (1975)

[13] Gurevich V., Volmen G., Teoriya razmernosti, IL, M., 1948

[14] Federer H., Geometric Measure Theory, Springer-Verlag, 1969 | MR

[15] Jonker J. B., “Immersions with semidefinite second fundamental forms”, Canad J. Math., 27:3 (1975), 610–617 | MR | Zbl

[16] Calabi E., Hartman P., “On the smoothness of isometries”, Duke Math. J., 37:4 (1970), 741–751 | DOI | MR

[17] Dmitriev V. G., “O podmnogoobraziyakh so vsyudu poluopredelennoi vtoroi osnovnoi formoi”, Zap. nauchn. sem. LOMI, 45 (1974), 68–70 | MR | Zbl

[18] Milka A. D., “Kratchaishie linii na vypuklykh poverkhnostyakh”, DAN SSSR, 248:1 (1979), 34–36 | MR | Zbl

[19] Milka A. D., “Metricheskoe stroenie odnogo klassa prostranstv, soderzhaschikh pryamye linii”, Ukr. geometr. sb., 4 (1967), 43–48 | MR | Zbl