In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $>3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,\alpha}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.
Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove that there are constants $\nu(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $\delta\geq 0$, \[ \mathrm{Ric}\geq-\delta^2, \qquad C_{m+1}\geq 1, \] then \[ \delta R\leq\nu(n,m) \quad\Longrightarrow\quad \mathrm{Vol} B_R(p) \leq C(n,m)R^m \quad \text{for every $p\in M$ and $R>0$.} \] In particular, taking $m=n-2$ and $\delta=0$ gives Gromov's conjectured codimension-two volume growth estimate under $\mathrm{Ric} \geq0$ and $\mathrm{Scal} \geq1$.
Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0<\gamma \leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\ge3$ and $\gamma>4(n-1)/(n-2)$, the space $R^\gamma(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2,"Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary.
G. Antonelli, Georg Frenck, Bernhard Hanke· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
