Metrics Of Positive Scalar Curvature Research Articles

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

Torsion Obstructions to Positive Scalar Curvature

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.

A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

We study notions of isotopy and concordance for Riemannian metrics on manifolds with boundary and, in particular, we introduce two variants of the concept of minimal concordance, the weaker one naturally arising when considering certain spaces of metrics defined by a suitable spectral ''stability'' condition. We develop some basic tools and obtain a rather complete picture in the case of surfaces.

Constrained deformations of positive scalar curvature metrics

Constrained deformations of positive scalar curvature metrics

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

Constrained deformations of positive scalar curvature metrics, II

AbstractWe prove that various spaces of constrained positive scalar curvature metrics on compact three‐manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean‐convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

Positive scalar curvature on manifolds with fibered singularities

Abstract A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ M_{\Sigma} with two strata: an open nonsingular stratum M ̊ \mathring{M} (a smooth open manifold) and a closed stratum β ⁢ M \beta M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M \beta M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ M_{\Sigma} with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k \mathbb{Z}/k or S 1 S^{1} , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin𝑐 line bundle.

Stability of the non–symmetric space E7/PSO(8)

We prove that the normal metric on the homogeneous space E7/PSO(8) is stable with respect to the Einstein-Hilbert action, thereby exhibiting the first known example of a non-symmetric metric of positive scalar curvature with this property.

A generalization of Geroch's conjecture

AbstractThe Theorem of Bonnet–Myers implies that manifolds with topology do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so‐called m‐intermediate curvature), and use stable weighted slicings to show that for and the manifolds do not admit a metric of positive m‐intermediate curvature.

Contractible 3-manifolds and positive scalar curvature (II)

In this article, we are interested in the question whether any complete contractible 3- manifold of positive scalar curvature is homeomorphic to \mathbb{R}^3 . We study the fundamental group at infinity, \pi_1^\infty , and its relationship to the existence of complete metrics of positive scalar curvature. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial \pi_1^\infty 1 is homeomorphic to \mathbb{R}^3 .

Nonnegative scalar curvature on manifolds with at least two ends

AbstractLet be an orientable connected ‐dimensional manifold with and let be a two‐sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's ‐bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin 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 2. We deduce as special cases that, if does not admit a metric of psc and , then does not carry a complete metric of psc and does not carry a complete metric of uniformly psc, provided that and , respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

A generalization of the Geroch conjecture with arbitrary ends

Using $\mu$-bubbles, we prove that for $3 \le n \le 7$, the connected sum of a Schoen-Yau-Schick $n$-manifold with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When either $3 \le n \le 5$, $1 \le m \le n-1$ or $6 \le n \le 7$, $m \in \{1, n-2, n-1\}$, we also show the connected sum $(M^{n-m}\times \mathbb{T}^m) \# X^n$ where $X$ is an arbitrary manifold does not admit a metric of positive $m$-intermediate curvature. Here $m$-intermediate curvature is a new notion of curvature introduced by Brendle, Hirsch and Johne interpolating between Ricci and scalar curvature.

Compacitification and Positive Mass Theorem for Fibered Euclidean End

We consider the Positive Mass Theorem for Riemannian manifolds (M^{n},g) with the asymptotic end (mathbb {R}^{k}times X^{n-k}, g_{mathbb {R}^{k}}+g_{X}) (kge 3) by studying the corresponding compactification problem. Here (X, g_X) is a compact scalar flat manifold. We show that the Positive Mass Theorem holds if certain generalized connected sum admits no metric of positive scalar curvature. Moreover we establish the rigidity result, namely, the mass is zero iff M is isometric to mathbb {R}^{k}times X^{n-k}.

On the homotopy type of the space of metrics of positive scalar curvature

Let M^d be a simply connected spin manifold of dimension d \geq 5 admitting Riemannian metrics of positive scalar curvature. Denote by \mathcal{R}^+(M^d) the space of such metrics on M^d . We show that \mathcal{R}^+(M^d) is homotopy equivalent to \mathcal{R}^+(S^d) , where S^d denotes the d -dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds M^d with d\geq 5 and d\neq 8 . In this case let W^d be the total space of the non-trivial S^{d-2} -bundle with structure group SO(d-1) over S^2 . Then \mathcal{R}^+(M^d) is homotopy equivalent to \mathcal{R}^+(W^d) .

Positive scalar curvature and homology cobordism invariants

AbstractWe give an obstruction to positive scalar curvature metrics on 4‐manifolds with the homology described in terms of homology cobordism invariants from Seiberg–Witten theory. The main tool of the proof is a relative Bauer–Furuta‐type invariant on a periodic‐end 4‐manifold.

Spaces of positive scalar curvature metrics on totally nonspin manifolds with spin boundary

In this article we study the space of positive scalar curvature metrics on totally nonspin manifolds with spin boundary. We prove that for such manifolds of certain dimensions, those spaces are not connected and have nontrivial fundamental group. Furthermore we show that a well-known propagation technique for detection results on spaces of positive scalar curvature metrics on spin manifolds ceases to work in the totally nonspin case.

A quantitative relative index theorem and Gromov’s conjectures on positive scalar curvature (with an appendix by Jinmin Wang and Zhizhang Xie)

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a \lambda -Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This \lambda -Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal \lambda -Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's cube inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov’s cube inequality on the bound of widths of Riemannian cubes and Gromov’s conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.

Families of diffeomorphisms and concordances detected by trivalent graphs

AbstractWe study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non‐trivial elements in homotopy groups are lifted to homotopy groups of the moduli space of ‐cobordisms . As a geometrical application, we show that those elements in for are also lifted to the rational homotopy groups of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups of moduli space of concordances of positive scalar curvature metrics on with fixed‐round metric on the boundary .

Interior Kasparov products for ϱ-classes on Riemannian foliated bundles

Let ι:F0→F1 be a suitably oriented inclusion of foliations over a manifold M, then we extend the construction of the lower shriek maps given by Hilsum and Skandalis to adiabatic deformation groupoid C*-algebras: we construct an asymptotic morphism (ιad[0,1))!∈En(C⁎(Gad[0,1)),C⁎(Gad[0,1))), where G and H are the monodromy groupoids associated with F0 and F1 respectively. Furthermore, we prove an interior Kasparov product formula for foliated ϱ-classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles.

On the space of riemannian metrics satisfying surgery stable curvature conditions

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.