Scalar Curvature Metrics Research Articles

The $S^3_{\boldsymbol{w}}$ Sasaki join construction

The main purpose of this work is to generalize the $S^{3}_{\boldsymbol{w}}$ Sasaki join construction $M \star_{\boldsymbol{l}} S^{3}_{\boldsymbol{w}}$ described in the authors' 2016 paper when the Sasakian structure on $M$ is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case ($c_{1}(\mathscr{D}) = 0$) we construct a polynomial whose coeffients are linear in the components of $\boldsymbol{w}$ and whose unique root in the interval $(1, \infty)$ completely determines the Sasaki–Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper of Apostolov and Calderbank as well as the relation with K-stability.

Dirac operators on foliations with invariant transverse measures

We extend the groundbreaking results of Gromov and Lawson, [13], to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian manifolds which admit invariant transverse measures. We prove a relative measured index theorem for pairs of such manifolds, foliations and operators, which are identified off compact subsets of the manifolds. We assume that the spectral projections of the leafwise operators for some interval [0,ϵ], ϵ>0, have finite dimensional images when paired with the invariant transverse measures. As a prime example, we show that if the zeroth order operators in the associated Bochner Identities are uniformly positive off compact subsets of the manifolds, then they satisfy the hypotheses of our relative measured index theorem. Using these results, we show that for a large collection of spin foliations, the space of positive scalar curvature metrics on each foliation has infinitely many path connected components.

Higher localised [formula omitted]-genera for proper actions and applications

For a finitely generated discrete group Γ acting properly on a spin manifold M, we formulate new topological obstructions to Γ-invariant metrics of positive scalar curvature on M that take into account the cohomology of the classifying space B_Γ for proper actions.In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher Aˆ-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of Γ-invariant positive scalar curvature on M. For classes arising from the subring of H⁎(B_Γ,R) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted L2-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of Wang-Wang [24] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of Mathai [17], which provided a partial answer to a conjecture of Gromov-Lawson on higher Aˆ-genera.If M is non-cocompact, we obtain obstructions to M being a partitioning hypersurface inside a non-cocompact Γ-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [12], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

The investigation of Riemannian metrics with lower scalar curvature bounds has been a central topic in differential geometry for decades. It addresses foundational problems, combining ideas and methods from global analysis, geometric topology, metric geometry and general relativity. Seminal contributions by Gromov during the last years have led to a significant increase of activities in the area which have produced a number of impressive results. Our workshop reflected the state of the art of this thriving field of research.

On the Dirac Spectrum of Homogeneous 3-Spheres

We show that any two left-invariant metrics on S^3cong {{text {SU}}}(2) which are isospectral for the associated classical Dirac operator D must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of D^2. We show analogous results for left-invariant metrics on {{text {SO}}}(3)=S^3/{pm 1} for each of its two spin structures.

The positive scalar curvature cobordism category

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside R+(Sd) if d≥6. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen’s Theorem B and instances of the Gromov–Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal’s theory of Γ-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen–Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes. A delooped version of the Atiyah–Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real K-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

Positive (p,n)-intermediate scalar curvature and cobordism

In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least 3 to a metric of positive scalar curvature which is a product near the boundary. We extend this construction for (p,n)-intermediate scalar curvature for 0≤p≤n−2 for surgeries in codimension at least p+3. We then use it to generalize a well known theorem of Carr. Letting Rsp,n>0(M) denote the space of positive (p,n)-intermediate scalar curvature metrics on an n-manifold M, we show for 0≤p≤2n−3 and n≥2, that for a closed, spin, (4n−1)-manifold M admitting a metric of positive (p,4n−1)-intermediate scalar curvature, Rsp,4n−1>0(M) has infinitely many path components.

A variant of Roe algebras for spaces with cylindrical ends with applications in relative higher index theory

In this paper, we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary. This paves the way for the classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.

Besse conjecture with positive isotropic curvature

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function f on the manifold that satisfies the following $$\begin{aligned} (1+f)\mathrm{Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. \end{aligned}$$ It has been conjectured that if (g, f) is a solution of the critical point equation, then g is Einstein and so (M, g) is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.

Sprays of weakly isotropic curvature

Spray geometry and Finsler geometry are related to each other because every Finsler metric on a manifold induces a spray on it. In the past decades, some sprays of scalar curvature and isotropic curvature are studied to tell the relations with Finsler metrics of scalar flag curvature and isotropic flag curvature. It is a natural way to study more new sprays with special curvature to find more interesting properties of sprays and their relations with Finsler metrics. In this paper, we introduce a new notion of curvature named weakly isotropic curvature which is more general than isotropic curvature for sprays. The relationship between this new curvature and the existed curvatures (Riemann curvature, χ-curvature and H-curvature) is revealed. Then we prove this curvature notion is a natural extension of weakly isotropic flag curvature for Finsler metrics. Two new groups of sprays of weakly isotropic curvature not induced by any Finsler metric are constructed based on our result.

On projective invariants of general spherically symmetric Finsler spaces in [formula omitted

In this paper we discuss projective invariants of general spherically symmetric Finsler metrics in Rn. We obtain the necessary and sufficient conditions for the metrics to be projectively Ricci flat, Weyl and W-quadratic types. In particular, we use the spray theory to give a short proof of the well-known theorem, that is, “Finsler manifold is of scalar flag curvature if and only if F is Weyl metric”. Therefore, considering the technique of the proof, we obtain a necessary and sufficient condition for the metrics of scalar flag curvature to be Weyl metric. Also, under a certain condition, we prove that projectively Ricci flat general spherically symmetric metrics coincide with the Douglas type metric.

Slant products on the Higson–Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.

A cohomological Seiberg–Witten invariant emerging from the adjunction inequality

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.

On scalar curvature of r-almost Yamabe solitons

The aim of this work is to study constant scalar curvature metrics of r-almost Yamabe solitons immersed into a Riemannian manifold. We deal with the compact and complete cases to obtain that an r-Yamabe soliton must have constant scalar curvature and some triviality results. Moreover, we also prove a Schur type inequality.

Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles

We study the existence of weighted extremal K\ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\ahler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal K\ahler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally K\ahler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For each admissible K\ahler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(-1,1)$, then there exists a weighted extremal K\ahler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no K\ahler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of K\ahler metrics of constant weighted scalar curvature in the rational admissible K\ahler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds

We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an end-periodic index theorem for the Dirac operator. As a result, we show that the bordism groups $\Omega^{spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.

Scalar positive immersions

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 [Formula: see text]-genus of Hitchin. When unobstructed we shall realize 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.

Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

Volume and macroscopic scalar curvature

We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of ell ^2-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

Some aspects of Ricci flow on the 4-sphere

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.