Scalar Curvature Metrics 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.

The Einstein-harmonic equations and constant scalar curvature Kähler metrics

Let (M,J) be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.

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.

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 .

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

A product model for generalizing Poincaré-type Kähler metrics

We begin by defining a type of Kähler metric near the zero section of a trivial holomorphic open disk bundle N N over a compact Kähler manifold X X by incorporating flows generated by holomorphic vector fields on X X . These metrics are then shown to deviate exponentially from Poincaré-type metrics on N ∖ X N\setminus X in terms of the log-polar distance from X X in N N . Lastly we see that they arise naturally when perturbing classes containing Poincaré-type Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class on X X does not admit a cscK metric.

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.

On isotropic Berwald scalar curvature

In this paper, we establish a closer relation between the Berwald scalar curvature and the [Formula: see text]-curvature. In fact, we prove that a Finsler metric has isotropic Berwald scalar curvature if and only if it has weakly isotropic [Formula: see text]-curvature. For Finsler metrics of scalar flag curvature and of weakly isotropic [Formula: see text]-curvature, they have almost isotropic [Formula: see text]-curvature if and only if the flag curvature is weakly isotropic.

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.

Optimal symplectic connections and deformations of holomorphic submersions

We give a general construction of extremal Kähler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kähler manifolds with constant scalar curvature, in a way that is compatible with the fibration structure. Thus we allow fibres that are K-semistable, rather than K-polystable; this is crucial to moduli theory. On these fibrations we phrase a partial differential equation whose solutions, called optimal symplectic connections, represent a canonical choice of a relatively Kähler metric. We expect this to be the most general construction of a canonical relatively Kähler metric provided all input is smooth. We use the notion of an optimal symplectic connection and the geometry related to it to construct Kähler metrics with constant scalar curvature and extremal metrics on the total space, in adiabatic classes.

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.

On modified Einstein tensors and two smooth invariants of compact manifolds

Let ( M , g ) (M,g) be a Riemannian n n -manifold, we denote by R i c Ric and S c a l Scal the Ricci and the scalar curvatures of g g . For each real number k > n k>n , the modified Einstein tensors denoted E i n k \mathrm {Ein}_k is defined to be E i n k ≔ S c a l g − k R i c \mathrm {Ein}_k ≔Scal\, g -kRic . Note that the usual Einstein tensor coincides with one half of E i n 2 \mathrm {Ein}_2 and E i n 0 = S c a l . g \mathrm {Ein}_0=Scal.g . It turns out that all these new modified tensors, for 0 > k > n 0>k>n , are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. The positivity of E i n k \mathrm {Ein}_k for some positive k k implies the positivity of all E i n l \mathrm {Ein}_l with 0 ≤ l ≤ k 0\leq l\leq k and so we define a smooth invariant E i n ( M ) \mathbf {Ein}(M) of M M to be the supremum of positive k’s that renders E i n k \mathrm {Ein}_k positive. By definition E i n ( M ) ∈ [ 0 , n ] \mathbf {Ein}(M)\in [0,n] , it is zero if and only if M M has no positive scalar curvature metrics and it is maximal equal to n n if M M possesses an Einstein metric with positive scalar curvature. In some sense, E i n ( M ) \mathbf {Ein}(M) measures how far M M is away from admitting an Einstein metric of positive scalar curvature. In this paper, we prove that E i n ( M ) ≥ 2 \mathbf {Ein}(M)\geq 2 , for any closed simply connected manifold M M of positive scalar curvature and with dimension ≥ 5 \geq 5 . Furthermore, for a compact 2 2 -connected manifold M M with dimension ≥ 6 \geq 6 and of positive scalar curvature, we show that E i n ( M ) ≥ 3 \mathbf {Ein}(M)\geq 3 . We demonstrate as well that the invariant E i n ( M ) \mathbf {Ein} (M) of a manifold M M increases after doing a surgery on M M or by assuming that M M has higher connectivity. We show that the condition E i n ( M ) ≤ n − 2 \mathbf {Ein}(M)\leq n-2 does not imply any restriction on the first fundamental group of M M . We define and prove similar properties for an analogous invariant namely e i n ( M ) \mathbf {ein}(M) . The paper contains several open questions.