Geodesic
Three-dimensional manifolds of nonnegative Ricci curvature, with boundary
Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 163-186 Cet article a éte moissonné depuis la source Math-Net.Ru

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A complete proof is given of the theorem, announced earlier, that a three-dimensional Riemannian manifold with nonnegative Ricci curvature and nonempty connected boundary of nonnegative mean curvature (or, more generally, with $H\geqslant0$ and $\operatorname{Ric}\geqslant-\min H^2$) is a handlebody (oriented or nonoriented). The proof uses the fact that subanalytic sets have finite triangulations and a generalized limit angle lemma; these enable one to control the reconstruction of the equidistants of the boundary. Figures: 3. Bibliography: 27 titles. 
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N. G. Ananov; Yu. D. Burago; V. A. Zalgaller. Three-dimensional manifolds of nonnegative Ricci curvature, with boundary. Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 163-186. http://geodesic.mathdoc.fr/item/SM_1987_56_1_a10/

[1] Ananov N. G., “Rimanovy mnogoobraziya s kraem ogranichennoi snizu srednei krivizny”, DAN SSSR, 249:6 (1979), 1289–1291 | MR | Zbl

[2] Ananov N. G., Burago Yu. D., Zalgaller V. A., Trekhmernye mnogoobraziya neotritsatelnoi krivizny Richchi, Preprint LOMI. R-3-83, Leningrad, 1983

[3] Bishop K., Krittenden K., Geometriya mnogoobrazii, Mir, M., 1967 | MR | Zbl

[4] Burago Yu. D., Zalgaller V. A., “Vypuklye mnozhestva v rimanovykh prostranstvakh neotritsatelnoi krivizny”, UMN, 32:3 (1977), 3–55 | MR | Zbl

[5] Massi U., Stollings Dzh., Algebraicheskaya topologiya. Vvedenie, Mir, M., 1977 | MR | Zbl

[6] Milnor Dzh., Teoriya Morsa, Mir, M., 1965 | MR

[7] Mostovskoi A. P., “Ob odnom obobschenii vypuklykh mnozhestv v rimanovom prostranstve”, Voprosy globalnoi geometrii. 31-e Gertsenovskie chteniya, Leningr. ped. in-t, 1978, 21–27 | MR

[8] Reshetnyak Yu. G., “Ob odnom obobschenii vypuklykh poverkhnostei”, Matem sb., 40(82) (1956), 381–388

[9] Aubin T., “Metriques riemanniennes et courbure”, J. Diff. Geom., 4 (1970), 383–424 | MR | Zbl

[10] Buchner M. A., “Simplicial structure of the real analitic cut locus”, Proc. Amer. Math. Soc., 64:1 (1977), 118–121 | DOI | MR | Zbl

[11] Cheeger J., Gromoll D., “The splitting theorem for manifolds of nonnegative Ricci courvature”, J. Diff. Geom., 6:1 (1971), 119–128 | MR | Zbl

[12] Cheeger J., Gromoll D., “On the structure of complete manifolds of nonnegative curvature”, Ann. Math., 96:3 (1972), 413–443 | DOI | MR | Zbl

[13] Dekster B. V., “Estimates of the length of a curve”, J. Diff. Geom., 12:1 (1977), 101–117 | MR | Zbl

[14] Ehrlich P. E., “Metric deformation of curvature. 1. Local convex deformation”, Geom. Dedic., 5:1 (1976), 1–23 | MR | Zbl

[15] Federer H., “Curvature measure”, Trans. Amer. Math. Soc., 93:3 (1959), 418–491 | DOI | MR | Zbl

[16] Green L., “Some open problems in differential geometry”, Proc. Symp. Pure Math., 27:1 (1975), 407–412 | MR

[17] Hamilton R. S., “Three-manifolds with positive Ricci curvature”, J. Diff. Geom., 17:2 (1982), 255–306 | MR | Zbl

[18] Hempel J., “3-manifolds”, Ann. Math. Stud., 1976, no. 86

[19] Hironaka H., “Subanalitic sets in number theory, algebraic geometry and commutative algebra”, Papers in honor of Y. Akizuki, Kinokuniya, Tokyo, 1977, 453–493 | MR

[20] Hironaka H., “Triangulations of algebraic sets”, Proc. Symp. Pure Math., 29 (1975), 165–185 | MR | Zbl

[21] Ichida R., “Riemannian manifolds with compact boundary”, Jokohama Math. J., 29:2 (1981), 169–177 | MR | Zbl

[22] Kasue A., “Ricci curvature, geodesics and some geodesic properties of Riemannian manifolds with boundary”, J. Math. Soc. Jap., 35:1 (1983), 117–131 | DOI | MR | Zbl

[23] Kleinjohan N., “Nächste Punkte in der Riemannschen Geometrie”, Math. Z., 176:3 (1981), 327–344 | DOI | MR | Zbl

[24] Lawson H. B. (Jr.), “Complete minimal surfaces in $S^3$”, Ann. Math., 92:2 (1970), 335–374 | DOI | MR | Zbl

[25] Meeks W., Simon L., Yau S. T., “Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature”, Ann. Math., 116:3 (1982), 621–659 | DOI | MR | Zbl

[26] Meeks W., Yau S. T., “Topology of three dimensional manifolds and the embedding problem in minimal surface theory”, Ann. Math., 112:3 (1980), 441–484 | DOI | MR | Zbl

[27] Schoen R., Yau S. T., “Complete three dimensional manifolds with positive Ricci curvature and scalar curvature”, Ann. Math. Stud., 1982, no. 102, 209–228 | MR | Zbl