TY - JOUR
T1 - Existence of polarized F-structures on collapsed manifolds with bounded curvature and diameter
AU - Cheeger, Jeff
AU - Rong, Xiaochun
PY - 1996
Y1 - 1996
N2 - We study the class of collapsed Riemannian n-manifolds with bounded sectional curvature and diameter. Our main result asserts that there is a constant. δ (n, d) > 0, such that if a compact n-manifold has bounded curvature, |K Mn| ≤ 1, bounded diameter, diam(Mn) ≤ d and sufficiently small volume, Vol(Mn) ≤ δ (n, d), then it admits a mixed polarized F-structure. As a consequence, infg Vol(Mn . g) = 0, where the infimum is taken over all metrics with |K (Mn, g) | ≤ 1. This assertion can be viewed as a weakened version of Gromov's "critical volume" conjecture.
AB - We study the class of collapsed Riemannian n-manifolds with bounded sectional curvature and diameter. Our main result asserts that there is a constant. δ (n, d) > 0, such that if a compact n-manifold has bounded curvature, |K Mn| ≤ 1, bounded diameter, diam(Mn) ≤ d and sufficiently small volume, Vol(Mn) ≤ δ (n, d), then it admits a mixed polarized F-structure. As a consequence, infg Vol(Mn . g) = 0, where the infimum is taken over all metrics with |K (Mn, g) | ≤ 1. This assertion can be viewed as a weakened version of Gromov's "critical volume" conjecture.
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U2 - 10.1007/BF02249258
DO - 10.1007/BF02249258
M3 - Article
AN - SCOPUS:0040203353
SN - 1016-443X
VL - 6
SP - 411
EP - 429
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 3
ER -