Codimension four regularity of generalized Einstein structures (Q6084221)

The authors prove a codimension 4 regularity theorem for non-collapsing sequences of bounded generalized Einstein structures and establish related bounds on various tensor fields, leading to some rigidity results for generalized Einstein manifolds. More precisely, let \((M,g)\) be an oriented Riemannian \(n\)-manifold and \(H=H_0+\dots +H_n\) be a smooth tensor field built up from \(k\)-forms \(H_k\) over \(M\). Moreover, define the symmetric \((0,2)\)-type tensor field \(H^2\) by \(H^2(X,Y):=g(H(X,\:\cdot\:),H(Y,\:\cdot\:))\), where the interior products such as \(H(X,\:\cdot\:)\) are regarded as sums of differential forms. Then \((M,g,H)\) is called a \textit{generalized Einstein manifold} if \[ \begin{cases} \mathrm{Ric}_g-\frac{1}{4}H^2 & =0,\\ \mathrm{d}H+Q_1(H) & =0,\\ \mathrm{d}^*_g+Q_2(H) & =0, \end{cases} \] where \(Q_i(H)\) are differential forms, which are quadratic in \(H\), defined using the metric \(g\), interior and wedge products. Introducing the \textit{generalized Ricci tensor} \(RC\) whose three direct sum components are the left hand sides of the three equations above respectively, the condition can also be written simply as \(RC=0\). Let \((M,g_j,H_j)\) be a sequence of generalized Einstein manifolds on a fixed \(n\)-manifold \(M\). Consider the induced sequence \((M, d_j,p_j)\) of pointed metric spaces, where \(p_j\in M\) are points satisfying that \(\mathrm{Vol}(B_1(p_j))>v>0\) (non-collapsing condition). Moreover, assume that the generalized Ricci tensors in this sequence satisfy the pointwise bounds \(\vert RC\vert_{g_j}\leq n-1\) along \(M\). The authors' main result is that the singular set \(S\) of the corresponding Gromov-Hausdorff limit \((X,d,p)\) satisfies \(\dim S\leq n-4\) (see Theorem 1.1 in the paper). In addition, effective volume estimates exist for this singular set, leading to \(L^2\) and \(L^p\) estimates for the original Riemannian curvature tensor of \(g\) and the tensor field \(H\) respectively (see Theorem 1.2 in the article). Finally, these estimates lead to various rigidity results concerning the diffeomorphism types of generalized Einstein spaces in \(4\) dimensions (see Corollary 1.3 in the article) and their standard Ricci-flatness (see Corollary 1.4 in the article). The appearance of four-dimensionality in this context is interesting because the motivation for the generalized Einstein condition comes from physics, more precisely it appears in certain supergravity theories and renormalization group flow equations. From the mathematical point of view, generalized Einstein structures are the critical points of the generalized Einstein-Hilbert action for generalized geometry.