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We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has psc outside a singular set of codimension $\geq 8$. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. We also discuss related phenomena on exotic spheres and tori. In addition, we provide examples of $\mathrm{L}^\infty$-metrics on $\mathbb{R}^n$ for certain $n \geq 8$ which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci-DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a $\mathrm{KO}$-theoretic condition which obstructs the existence of $\mathrm{L}^\infty$-metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics., Comment: 16 pages, 2 figures; v2: added new examples on exotic spheres and tori, miscellaneous minor improvements

We establish new mean curvature rigidity theorems of spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold which holds even when the scalar curvature is not assumed to be non-negative., Comment: 22 pages; v2: additional result in section 3 and miscellaneous minor improvements

Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $\mu$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow., Comment: Improved exposition. Comments are welcome!

We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu-|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e. $\mathrm{E} < |\mathrm{P}|$), there exists a constant $R>0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten's proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle., Comment: 18 pages

Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "Four lectures". Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles. We argue that the existence of a non-zero harmonic spinor field forces $f$ to be quasiregular in the sense of Reshetnyak, and in this way connect the powerful theory for quasiregular maps to the Atiyah-Singer index theorem., Comment: 30 pages, minor changes and corrections, bibliography updated, to appear in J. Eur. Math. Soc

Let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and let $Y\subset M$ be a two-sided closed connected incompressible hypersurface which does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Using Gromov's $\mu$-bubbles, we show that $M$ does not admit a complete metric of psc. We provide an example showing that the spin/non-spin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension $7$, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension two. We deduce as special cases that, if $Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M := Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times \mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that $\dim(M) \leq 7$ and $\dim(N) \leq 7$, respectively. This solves, up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds., Comment: 19 pages; v2: minor improvements. To appear in J. Topol

Let $\mathcal{E}$ be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that if $\mathcal{E}$ has negative ADM-mass, then there exists a constant $R > 0$, depending only on $\mathcal{E}$, such that $M$ must become incomplete or have a point of negative scalar curvature in the $R$-neighborhood around $\mathcal{E}$ in $M$. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions $\leq 7$ assuming Schwarzschild asymptotics on the end $\mathcal{E}$. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end $\mathcal{E}$. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature., Comment: 20 pages; v2: update discussion in introduction

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing $\hat{\mathrm{A}}$-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions., Comment: 48 pages, 2 figures; v2: optimality of long neck problem and improvements of exposition; v3: minor improvements and added two figures. To appear in Geometry & Topology

In this paper, we define a relative $L^2$-$\rho$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a $4k+3$-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in turn implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised $\eta$-invariants for odd-dimensional spin manifolds with boundary, which could then be used to obtain similar results for $4k+1$-dimensional manifolds., Comment: 18 pages

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a "long neck principle" for a compact Riemannian spin $n$-manifold with boundary $X$, stating that if $\textrm{scal}(X)\geq n(n-1)$ and there is a nonzero degree map into the sphere $f\colon X\to S^n$ which is strictly area decreasing, then the distance between the support of $\textrm{d} f$ and the boundary of $X$ is at most $\pi/n$. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold $X$ obtained by removing $k$ pairwise disjoint embedded $n$-balls from a closed spin $n$-manifold $Y$. We show that if $\textrm{scal}(X)>\sigma>0$ and $Y$ satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $\partial X$ is at most $\pi \sqrt{(n-1)/(n\sigma)}$. Finally, we consider the case of a Riemannian $n$-manifold $V$ diffeomorphic to $N\times [-1,1]$, with $N$ a closed spin manifold with nonvanishing Rosenberg index. In this case, we show that if $\textrm{scal}(V)\geq\sigma>0$, then the distance between the boundary components of $V$ is at most $2\pi \sqrt{(n-1)/(n\sigma)}$. This last constant is sharp by an argument due to Gromov., Comment: Revised version to appear in GAFA. In Theorem A, the map f is strictly area decreasing, but see Remark 1.7. Technical issues fixed in the proofs of Lemma 5.5 and Theorem 5.2. The inequality in Proposition 3.8 is slightly strengthened

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f., Comment: 13 pages. v2: minor corrections, replaced lemma 4.3 by a reference. v3: accepted for publication in Proc.AMS, version adapted to journal layout

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schr\"{o}dinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert $C^\ast$-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension-one., Comment: Revised version. The material from Section 3 of the old version has been moved to the appendix

We give a short proof of the following fact. Let $\Sigma$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $\Sigma$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that if $\Sigma$ admits a complete metric whose Gaussian curvature is nonnegative and positive at one point, then $\Sigma$ is diffeomorphic to $\mathbb{R}^2$., Comment: Proposition 2.16 of the old version is made the main theorem

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's $L^2$-index theorem, which states that the index of a Callias-type operator on a non-compact manifold $M$ is equal to the $\Gamma$-index of its lift to a Galois cover of $M$. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds., Comment: 35 pages

We provide criteria for self-adjointness and {\tau}-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace {\tau}. We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations., Comment: 15 pages. Final version, to appear in Journal of Noncommutative Geometry

We study the minimizer u of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense., Comment: 1 figure