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We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities.

    Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. 

    Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components. 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove Gromov's conjecture on the total mean curvature of fill-ins in various cases.

Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. 

For spin fill-ins, where we permit the mean curvature to take negative values, we build on a recent positive mass theorem with creases by Kazaras--Khuri--Lin.

For non-spin fill-ins of spin manifolds, where we assume the mean curvature to be non-negative, we develop a novel quantitative surgery process to reduce the general situation to a result of Shi--Wang--Wei.

We also treat the case of fill-ins of non-spin manifolds, provided there is a fixed positive lower bound on the mean curvature.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove  Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-like singularities and Lipschitz comparison maps to spheres. We use the  language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest. Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we  generalize a  Lipschitz rigidity  result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Bär using an upper estimate for the smallest Dirac eigenvalue.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds without boundary, we assume that the Lie group contains a normal S1-subgroup with fixed-point components of codimension two. In this situation, the existence of invariant metrics of positive scalar curvature was known previously.

For the proof, we combine equivariant Morse theory with conformal deformations near unstable manifolds. On manifolds without boundary, we also use local flexibility properties of positive scalar curvature metrics and the smoothing of mean-convex singularities.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped  with   strictly log-concave warping functions. This generalizes earlier results of  Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least $3$ minus two antipodal points, thus resolving a problem in Gromov's  ``Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics. 

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove the following Lipschitz rigidity result in scalar curvature geometry. 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, and on the theory of quasiregular maps due to Reshetnyak.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II58Profinite perspectives on l2-cohomology73Geometric Chern characters in p-adic equivariant K-theory

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.

The main application is a general approximation result by sections which have very restrictive local properties an open dense subsets. This shows, for instance, that given any K∈R every manifold of dimension at least two carries a complete C^1,1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course this is impossible for C^2-metrics in general.

Related project(s):
5Index theory on Lorentzian manifolds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.

Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimensions at least five which have odd order abelian fundamental groups, are non-spin and  atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Based on  Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of  spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over  the sphere bundle $\mathbb{S}( \mathbb{R} \oplus V)$  in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive
scalar curvature metrics on closed simply connected manifolds in dimensions at least 
five appears on spin manifolds, and is given by the non-vanishing of the \(\alpha\)-genus 
of Hitchin. 
When unobstructed we will in this paper realise  a positive scalar curvature metric by an 
immersion into Euclidean space whose dimension is uniformly close to the classical Whitney 
upper-bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure 
for constructing positive scalar curvature metrics.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

The canonical map from the \(\mathbb{Z}/2\)-equivariant Lazard ring to the \(\mathbb{Z}/2\)-equivariant complex bordism ring is an isomorphism. 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds