Regularity and compactness of harmonic-Einstein equations (Q1709760)

The authors consider a pointed Riemannian manifold \((M,x,g)\) with \(\mathrm{Vol}(B_1(x))\geq v>0\) that satisfies the harmonic-Einstein equation \(\operatorname{Ric}g-\nabla u\otimes \nabla u=\lambda g\) with \(|\lambda|\leq n-1\), where \(u:(M,g)\to (N,h)\) is a harmonic map to a fixed compact Riemannian manifold \((N,h)\). For any \(p<1\), they prove the \(L^ p\) curvature estimate \(\int_{B_1 (x)} |Rm|^p \leq C(p,N,v)\). As a consequence, if \((N,h)\) has nonpositive sectional curvature, then \(|\operatorname{Ric} |\leq C(n,v,N)\). This means that the harmonic-Einstein equation automatically implies bounded Ricci curvature provided nonpositivity of the sectional curvature of \((N,h)\).