Poincaré inequalities for maps with target manifold of negative curvature

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.