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Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

New extremal problems for the Riemannian recognition program via Alexandrov geometry

Author(s): Karsten Grove; Steen Markvorsen
Journal: J. Amer. Math. Soc. 8 (1995), 1-28.
MSC: Primary 53C23; Secondary 53C20
MathSciNet review: 1276824
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References:

[AG]
U. Abresch and D. Gromoll, On complete manifolds with non-negative Ricci curvature, J. Amer. Math. Soc. 3 (1990), 355-374. MR 1030656 (91a:53071)

[Be]
V. N. Berestowski, Spaces with bounded curvature and distance geometry, Sibirsk. Mat. Zh. 27 (1986), 26-34. MR 847410 (88e:53110)

[BGP]
Y. Burago, M. Gromov, and G. Perelman, A.D. Alexandrov's spaces with curvatures bounded from below I, Uspekhi Mat. Nauk 47 (1992), 3-51; English transl., Russian Math. Surveys 47 (1992), 1-58. MR 1185284 (93m:53035)

[CE]
J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland Math. Library, vol. 9, North-Holland, Amsterdam and New York, 1975. MR 0458335 (56:16538)

[C]
H. S. M. Coxeter, Introduction to geometry, Wiley, New York, 1961. MR 0123930 (23:A1251)

[F]
I. Fary, Sur la courbure totale d'une courbe gauche faisant un naeud, Bull. Soc. Math. France 77 (1949), 128-138. MR 0033118 (11:393h)

[GG]
D. Gromoll and K. Grove, A generalization of Berger's rigidity theorem for positively curved manifolds, Ann. Sci. École. Norm. Sup. 20 (1987), 277-239. MR 911756 (88k:53062)

[G1]
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. MR 623534 (83b:53041)

[G2]
-, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1-148. MR 697984 (85h:53029)

[G]
K. Grove, Metric and topological measurements of manifolds, Proc. Internat. Congr. Math. (Kyoto, Japan, 1990), Math. Soc. Japan and Springer-Verlag, Berlin and New York, 1991, pp. 511-519. MR 1159238 (93c:53030)

[GM]
K. Grove and S. Markvorsen, Curvature, triameter and beyond, Bull. Amer. Math. Soc. (N. S.) 27 (1992), 261-265. MR 1152160 (93b:53036)

[GP1]
K. Grove and P. Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), 195-206. MR 951512 (90a:53044)

[GP2]
-, Homotopy types of positively curved manifolds with large volume, Amer. J. Math 110 (1988), 1183-1188. MR 970124 (89m:53063)

[GP3]
-, Manifolds near the boundary of existence, J. Differential Geom. 33 (1991), 379-394. MR 1094462 (92a:53067)

[GP4]
-, A pinching theorem for homotopy spheres, J. Amer. Math. Soc. 3 (1990), 671-677; On the excess of metric spaces and manifolds, preprint UMCP, 1989. MR 1049696 (91e:53040)

[GP5]
-, Volume comparison à la Alexandrov, Acta Math. 169 (1992), 131-151. MR 1179015 (93j:53057)

[GP6]
K. Grove and P. Petersen, A radius sphere theorem, Invent. Math. 112 (1993), 577-583. MR 1218324 (94e:53034)

[GPW]
K. Grove, P. Petersen V, and J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), 205-213; Erratum, Invent. Math. 104 (1991), 221-222. MR 1029396 (90k:53075)

[GW]
K. Grove and F. Wilhelm, Hard and soft packing radius theorems, preprint. MR 1343322 (96h:53054)

[H]
R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306. MR 664497 (84a:53050)

[HK]
W.-Y. Hsiang and B. Kleiner, On the topology of positively curved $ 4$-manifolds with symmetry, J. Differential Geom. 29 (1989), 615-621. MR 992332 (90e:53053)

[N]
F. Nielsen, On the sum of distances between $ n$ points on a sphere, Nordisk Mat. Tidsskr. 13 (1965), 45-50. (Danish)

[OSY]
Y. Otsu, K. Shiohama, and T. Yamaguchi, A new version of differentiable sphere theorem, Invent. Math. 98 (1989), 219-228. MR 1016261 (90i:53049)

[P]
G. Perelman, Alexandrov's spaces with curvatures bounded from below II, preprint.

[Pt1]
P. Petersen V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), 387-395. MR 1037407 (91d:53070)

[Pt2]
-, Gromov-Hausdorff convergence of metric spaces, Proc. Sympos. Pure Math., vol. 54, Part 3, Amer. Math. Soc., Providence, RI, 1993, pp. 489-504. MR 1216641 (94b:53079)

[Pl]
C. Plaut, Spaces of Wald curvature bounded below, J. Geom. Anal. (to appear). MR 1402389 (97j:53043)

[R]
W. Rinow, Die Innere Geometrie der Metrischen Räume, Springer-Verlag, Berlin and New York, 1961. MR 0123969 (23:A1290)

[Y]
T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), 317-357. MR 1097241 (92b:53067)

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Additional Information:

DOI: 10.1090/S0894-0347-1995-1276824-4
PII: S0894-0347-1995-1276824-4
Copyright of article: Copyright 1995, American Mathematical Society




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