Metrics Of Positive Scalar Curvature Research Articles

Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets

We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $\Omega^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $\pi_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in $\pi_* Diff(D^m,\partial)$ for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in $\pi_* Diff(M)$ where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of $(PL/O)_{(p)}$ for the prime 2; (f) for any $bP_{8j+4}$-sphere (where j > 0) of order which divides 4, the corresponding element in $\pi_0 Diff(D^{8j+2},\partial)$ lifts to $\pi_{8j-4} Diff(D^6,\partial)$, i.e., lies correspondingly deep down in the Gromoll filtration.

Enlargeability, foliations, and positive scalar curvature

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes’ vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes.

Two-cocycle twists and Atiyah–Patodi–Singer index theory

AbstractWe construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian manifolds with positive scalar curvature play an important role in mathematics and general relativity. Obstruction and existence results are connected to index theory, bordism theory and homotopy theory, using methods from partial differential equations and functional analysis. The workshop led to a lively interaction between mathematicians working in these areas.

Moduli spaces of invariant metrics of positive scalar curvature on quasitoric manifolds

We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are nontrivial.

A splitting theorem for the Seiberg-Witten invariant of a homology S1× S3

We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Fr{\o}yshov invariant $h(X)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.

Rigidity of outermost MOTS: the initial data version

In [5], a rigidity result was obtained for outermost marginally outer trapped surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This allowed one to treat the "borderline case" in the author's work with R. Schoen concerning the topology of higher dimensional black holes [8]. The proof of this rigidity result involved bending the initial data manifold in the vicinity of the MOTS within the ambient spacetime. In this note we show how to circumvent this step, and thereby obtain a pure initial data version of this rigidity result and its consequence concerning the topology of black holes.

On the geometry and topology of initial data sets with horizons

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set $(M, g, K)$ such that the boundary $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions $3\leq n \leq 7$, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary $\partial M$ induced by $g$. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem \cite{GS06} to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in \cite{EGP13} in dimension $n=3$, to higher dimensions.

Minimal hypersurfaces and bordism of positive scalar curvature metrics

Let (Y, g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen–Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume $$(Y_0,g_0)$$ and $$(Y_1,g_1)$$ are closed psc-manifolds equipped with stable minimal hypersurfaces $$X_0 \subset Y_0$$ and $$X_1\subset Y_1$$ . Under natural topological conditions, we show that a psc-bordism $$(Z,{\bar{g}}) : (Y_0,g_0)\rightsquigarrow (Y_1,g_1)$$ gives rise to a psc-bordism between $$X_0$$ and $$X_1$$ equipped with the psc-metrics given by the Schoen–Yau construction.

Generic finiteness of minimal surfaces with bounded Morse index

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by B. White) the same conclusion holds true for any closed 3-manifold.

Wrong way maps in uniformly finite homology and homology of groups

Given a non-compact Riemannian manifold M and a submanifold $$N \hookrightarrow M$$ of codimension q, we will construct under certain assumptions on both M and N a wrong way map $$H^\mathrm {uf}_*(M) \rightarrow H^\mathrm {uf}_{*-q}(N)$$ in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct a map $$H_*(B\pi _1 M) \rightarrow H_{*-q}(B\pi _1 N)$$ . As applications we discuss obstructions to the existence of positive scalar curvature metrics and to inessentialness.

On Higher Graph Manifolds

In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and show that they do not admit positive scalar curvature metrics. In the case without any pure pieces, we show the Yamabe invariant vanishes.

The two definitions of the index difference

Given two metrics of positive scalar curvature on a closed spin manifold, there is a secondary index invariant in real K K -theory. There exist two definitions of this invariant: one of a homotopical flavor, the other one defined by an index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this result to the case of two families of positive scalar curvature metrics, parametrized by a finite CW complex. In essence, we prove a generalization of the classical “spectral-flow-index theorem” to the case of families of real operators.

Positive scalar curvature on foliations

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the famous theorem of Schoen-Yau and Gromov-Lawson on the non-existence of metrics of positive scalar curvature on torus to the case of foliations. Moreover, our method, which is partly inspired by the analytic localization techniques of Bismut-Lebeau, also applies to give a new proof of the celebrated Connes vanishing theorem without using noncommutative geometry.

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$-orientation from the infinite loop space of the Madsen--Tillmann--Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$-dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov--Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah--Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral-flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen--Weiss theorem, proven by Galatius and the third named author.

Aspects of Positive Scalar Curvature and Topology I

This is the second and concluding part of a survey article. Whether or not a smooth manifold admits a Riemannian metric whose scalar curvature function is strictly positive is a problem which has been extensively studied by geometers and topologists alike. More recently, attention has shifted to another intriguing problem. Given a smooth manifold which admits metrics of positive scalar curvature, what can we say about the topology of the space of such metrics? We provide a brief survey, aimed at the non-expert, which is intended to provide a gentle introduction to some of the work done on these deep questions.

Higher rho invariants and the moduli space of positive scalar curvature metrics

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group P˜(M) of positive scalar curvature metrics is defined to be the quotient abelian group of this action, i.e. the coinvariant of the action. P˜(M) measures the size of the moduli space of positive scalar curvature metrics on M. In this paper, we use the higher rho invariant and the finite part of the K-theory of the group C⁎-algebra of π1(M) to give a lower bound of the rank of the moduli group P˜(M). The key idea of our proof is the use of higher rho invariant; this is a secondary invariant associated to Dirac operators and in particular depends on the choice of Riemannian metric. We show that the higher rho invariant remains unchanged in a certain K-theory group under the action of the diffeomorphism group, allowing us to distinguish elements in P˜(M).

Metrics of positive scalar curvature and unbounded min-max widths

In this work, we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to 6 and arbitrarily large widths. The search for metrics with such properties is motivated by the rigidity result of min-max minimal spheres in three-manifolds obtained by Marques and Neves (Duke Math J 161(14):2725–2752, 2012).

The Index of Dirac Operators on Incomplete Edge Spaces

We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a "geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.

Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces

This paper produces explicit strongly Hermitian Einstein–Maxwell solutions on the smooth compact 4-manifolds that are $$S^2$$ -bundles over compact Riemann surfaces of any genus. This generalizes the existence results by LeBrun (J Geom Phys 91:163–171, 2015, The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016). Moreover, by calculating the (normalized) Einstein–Hilbert functional of our examples, we generalize Theorem E of LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), which speaks to the abundance of Hermitian Einstein–Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein–Maxwell solutions, first found in LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.