A theorem on the tension field

Abstract:

Suppose $M$ and $N$ are complete Riemannian manifolds; $M$ with Ricci curvature bounded from below and $N$ with sectional curvature bounded from above by a constant ${K_0}$. Let $f:M \to N$ be a smooth map such that $f(M) \subset {B_R}$, where ${B_R}$ is a normal ball in $N$ and furthermore $R < \pi /2\sqrt {{K_0}}$ if ${K_0} > 0$. If the energy density $e(f)$ is bounded below by a positive constant, then there is a point $P \in M$ such that the tension field $\tau (f)$ at $P$ is different from zero.