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Poincaré inequalities for maps with target manifold of negative curvature


Kappeler, T; Schroeder, Viktor; Kuksin, S (2005). Poincaré inequalities for maps with target manifold of negative curvature. Moscow Mathematical Journal, 5(2):399-414, 494.

Abstract

We prove that for any given homotopic C1-maps u, v: G → M in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between u and v can be bounded by 3(length(u) + length(v)) + C(κ, ρ/20), where ρ>0 is a lower bound of the injectivity radius and −κ<0 an upper bound for the sectional curvature of M. The constant C(κ, ε) is given by

C(κ, ε) = 8 sh−1κ(1) + 8 sh−1κ (1/shκ(ε))

with shκ(t) = sinh(√{κ} t). Various applications are given.

Abstract

We prove that for any given homotopic C1-maps u, v: G → M in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between u and v can be bounded by 3(length(u) + length(v)) + C(κ, ρ/20), where ρ>0 is a lower bound of the injectivity radius and −κ<0 an upper bound for the sectional curvature of M. The constant C(κ, ε) is given by

C(κ, ε) = 8 sh−1κ(1) + 8 sh−1κ (1/shκ(ε))

with shκ(t) = sinh(√{κ} t). Various applications are given.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Negative sectional curvature, short homotopies, Poincaré inequality.
Language:English
Date:2005
Deposited On:03 Mar 2010 13:44
Last Modified:19 Feb 2018 19:59
Publisher:Independent University of Moscow
ISSN:1609-3321
OA Status:Green
Official URL:http://www.ams.org/distribution/mmj/vol5-2-2005/abst5-2-2005.html#kappeler-etal

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