Rank zero and rank one sets of harmonic maps (Q1601133)

This paper studies the sizes of the sets of points where a harmonic map from a ball in \(\mathbb{R}^n\) to a smooth compact Riemannian manifold \(N\) is of rank zero or of rank one. This generalizes the question of nonsingular nodal sets and critical nodal sets which have been studied intensively for harmonic functions, and, more generally, for solutions of elliptic equations of the second order. The results are as follows. If \(u\) is either a smooth or a minimizing harmonic map from the ball to \(N\), then the rank zero set \(R_0(u)\) is countably \((n{-}2)\)-rectifiable unless \(u\) maps the whole ball to a point in \(N\). Also, the rank one set \(R_1(u)\) is countably \((n{-}1)\)-rectifiable unless \(u\) maps the whole ball to a point or a geodesic in \(N\). There are bounds for the corresponding Hausdorff measures of \(R_0(u)\) and \(R_1(u)\), at least if intersected with the ball of half size, which depend on the manifold \(N\), the supnorm of the gradient of \(u\) (in the smooth case) or the total energy of \(u\), and some term which controls the oscillation of \(u\) (in the rank zero case) or the deviation of \(u\) from geodesic maps (in the rank one case).