ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 35,49,53,58 Tachikawa, Atsushi, Science University of Tokyo, Japan A Nonexistence Result of Harmonic Maps between Hadamard manifolds Let N = (N^{n}, g) be a Hadamard manifold i. e. a simply connected complete Riemannian manifold with non positive sectional curvatures. For some fixed q_{0} ∈ N, let σ(q_{0}, q)(t) be the geodesic curve such that σ(q_{0}, q)(0) = q_{0} and σ(q_{0}, q)(1) = q. Let us define k_{q_{0}}(q) and λ_{q_{0}}(q) as : k_{q_{0}}(q) := max{K(q ; π) :

π ∋ σ′(q_{0}, q)(1)\}, λ_{q_{0}}(q) := inf \{\frac{∥ξ∥_{g(q)}^{2}}{|ξ|^{2}} ; ξ · σ′(q_{0}, q)(1) = 0\} , where

K(q ; π) denotes the sectional curvature of N at q with respect to a plane section π. The following result will be presented. Theorem Let N = (N^{n}, g) be a Hadamard n-manifold which satisfies

{dist(q_{0}, q)}^{2}|k_{q_{0}}(q)| ≥ κ > 0 as dist(q_{0}, q) → ∞.

for some q_{0} ∈ N. Then there exists no harmonic map U : ℝ^{m} → N which satisfies the following nondeneracy condition at infinity:

lim inf_{|x|→∞} (\frac{|x|^{2}}{ρ^{2}λ_{q_{0}}(U(x))}) (e(U)(x) − e(ρ)(x)) > 0,

where e(U)(x) denotes the energy density of U at x, ρ(x) = dist_{N}(U(x), q_{0}),

and e(ρ)(x) = \frac{1}{2} ∑_{α=1}^{m} |\frac{∂ρ}{∂x^{α}} |^{2}.