Regularity and compactness of harmonic-Einstein equations

Let (M, x, g) be pointed Riemannian manifold with \(\mathrm{Vol}(B_1(x))\ge \mathrm{v}>0\) and satisfy harmonic-Einstein equation \(\mathrm{Ric}_g-\nabla u\otimes \nabla u=\lambda g\) with \(|\lambda |\le n-1\), where \(u:(M,g)\rightarrow (N,h)\) is a harmonic map to a fixed compact Riemannian manifold (N, h). Then for any \(p<1\), we prove the \(L^p\) curvature estimate Open image in new window . As a consequence, if (N, h) has nonpositive sectional curvature, we have \(|\mathrm{Ric}|\le C(n,\mathrm{v},N)\). That means harmonic-Einstein equation automatically implies bounded Ricci curvature provided nonpositive sectional curvature of (N, h). Let \((X,x_\infty , d,u_\infty )\) be the limit of a sequence of harmonic-Einstein manifolds \((M_i,x_i,g_i,u_i)\). We show that the singular set of X is closed and the convergence is smooth on the regular part. We also prove an orbifold type compactness theorem of harmonic-Einstein equations with bounded \(\int _M|\mathrm{Rm}|^{n/2}\) without assuming nonpositive sectional curvature of (N, h), which generalizes Xu’s compactness result in Yiyan (Adv. Math. 231(2):680–708, 2012).