Quantitative stratification and the regularity of harmonic maps and minimal currents

We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m $ between Riemannian manifolds include a priori bounds in $ W^{1,p} \cap W^{2,{p}/{2}} $ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the L_p norm of derivatives. Namely, we obtain L_p bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori L_p bounds for the second fundamental form A for all p < 7. Previously known bounds were for $ p < 4+ \sqrt{{8}/{n}} $ in the smooth immersed stable case. Again, the full theorem is much stronger and yields L_p bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem.