The heat flow on manifolds. Existence and uniqueness of harmonic maps into nonpositively curved image manifolds

Abstract

In order to warm up, we shall first present the linear case, i.e. look at

$$ \frac{{\partial \alpha \left( {x,t} \right)}}{{\partial t}} - {\Delta ^ - }\alpha \left( {x,t} \right) = 0 $$

((3.1.1))

$$ \alpha \left( {x,0} \right) = {\alpha _0}\left( x \right) $$

((3.1.2))

where $0 \leqslant t < \infty ,x \in N,$ , where N is a compact Riemannian manifold of dimension n, and α(·, t) and α ₀ are k-formson N.