Positive Scalar Curvature Metrics

• Positive scalar curvature metrics are Riemannian metrics with strictly positive scalar curvature at every point, linking geometry with topology and index theory.
• They are analyzed via surgery techniques and index-theoretic obstructions, enabling classification and moduli space studies especially on spin manifolds.
• Recent research uncovers rich moduli topology with infinite homotopy groups and group-theoretic invariants, influencing practical construction methods.

Positive scalar curvature metrics are a central object of study in differential geometry, global analysis, and geometric topology. On a given smooth manifold, the existence, classification, and moduli of such metrics probe deep connections between geometry, topology, index theory, and even areas of mathematical physics. This article presents foundational definitions, main classification theorems, index-theoretic obstructions, detailed moduli space topology, new phenomena for manifolds with boundary and singularities, as well as group-theoretic and analytic invariants.

1. Definitions, Key Properties, and Classification Principles

A Riemannian metric $g$ on an $n$-dimensional manifold $M$ is said to have positive scalar curvature (psc) if the scalar curvature $R_g(p)>0$ at every point $p\in M$. For closed manifolds, the set of psc metrics $\Riem^+(M)$ is open in the Fréchet manifold of all smooth metrics, and forms an infinite-dimensional space typically with rich topology (Schick, 2014).

Key notions:

• psc-isotopy: A smooth path $g_t$ ($t\in[0,1]$) in $\Riem^+(M)$ between $g_0$ and $g_1$ (Botvinnik, 2012).
• psc-concordance: A psc metric $\bar{g}$ on $M\times[0,1]$ restricting near the boundary to product metrics $g_0+dt^2$, $g_1+dt^2$ (Botvinnik, 2012).
• Moduli Spaces: Quotients $\Riem^+(M)/\Diff(M)$ under diffeomorphism action, measuring equivalence classes of psc metrics, potentially with singular or boundary conditions (Schick, 2014, Xie et al., 2013).
• Bordism and Concordance Groups: Classification up to bordism (psc metrics on all manifolds mapping to $M$ modulo psc bordism) and up to concordance (psc metrics on $M$ modulo cylinder extension), naturally linked to spin bordism and index theory (Schick et al., 2020).

For 3-manifolds with boundary, metric and mean curvature interplay critically: $H_g>0$ characterizes mean-convex boundaries; $H_g=0$ minimal boundary; full topological classification is known (Carlotto et al., 2019). Contractibility and path-connectedness results for moduli spaces of psc metrics under various geometric constraints appear throughout the literature (Baer et al., 20 Mar 2025, Carlotto et al., 2021).

2. Index-Theoretic Obstructions and Analytic Classification

The existence of psc metrics is fundamentally obstructed by index theory. For closed spin manifolds:

• Lichnerowicz–Hirzebruch Obstruction: If $M^n$ is spin and admits a psc metric, then $\AHat(M)=0$ (Schick, 2014).
• Rosenberg Index: In presence of fundamental group $\Gamma=\pi_1(M)$, the (twisted) Dirac operator index $\alpha(M)\in KO_n(C^*\Gamma)$ must vanish (Schick, 2014, Pennig, 2011).
• Twisted K-theory Obstructions: For manifolds with spin universal cover but not necessarily globally spin, twisted index obstructions $\theta(M)\in K_0(C^*_{\max}(\widehat\pi\to\pi))$ must vanish (Pennig, 2011).

When these invariants are nonzero, no psc metric exists. The generalized Gromov–Lawson–Rosenberg conjecture connects spin bordism and index-theory as the "only" obstructions, though counterexamples are known.

3. Surgery, Concordance, and Isotopy Results

A decisive tool in constructing psc metrics on high-dimensional simply-connected manifolds is surgery theory:

• Gromov–Lawson Surgery: Surgeries in codimension $\ge 3$ preserve the existence of psc metrics, allowing one to reduce to handlebodies and prime factors supporting known metrics (Botvinnik, 2012, Schick, 2014).
• Stolz’s Classification: For closed simply-connected spin manifolds, $n\ge5$, $M$ admits a psc metric if and only if $\AHat(M)=0$ (Schick, 2014, Florit et al., 2019).
• Concordance vs. Isotopy: For simply connected $M^n$ ($n\geq5$), two psc metrics are psc-concordant if and only if they are psc-isotopic up to a boundary diffeomorphism (Botvinnik, 2012). The proof involves Gromov–Lawson surgery, the Cerf–Igusa–Hatcher pseudo-isotopy theorem, conformal Laplacian deformation, and Ricci flow smoothing.

Topological classification results for psc metrics with curvature constraints on boundary (mean-convex or minimal) use doubling, smoothing of corners (Miao’s approach), and Ricci flow with surgery (Carlotto et al., 2019, Baer et al., 2020).

4. Topology of the Space and Moduli of PSC Metrics

Recent advances reveal highly nontrivial topology of $\Riem^+(M)$ and its moduli spaces:

• Infinite Order Homotopy Groups: Families of psc metrics distinguished by higher index invariants provide elements of infinite order in various homotopy groups of $\Riem^+(M)$ (Hanke et al., 2012, Crowley et al., 2012, Wiemeler, 2016).
• Moduli Groups and Concordance Classes: The abelian group $P(M)$ of concordance classes of psc metrics (Grothendieck group), and its further quotient by diffeomorphism action, are computed in terms of analytic and group-theoretic data, e.g., finite part $K$-theory of group $C^*$-algebras, index difference, and higher rho invariants (Xie et al., 2013, Schick et al., 2020).
• Contractibility under Symmetry: For many manifolds with symmetry (e.g., torus manifolds with $S^1$ or $T^n$ action), spaces of invariant psc metrics are contractible (Baer et al., 20 Mar 2025).
• Boundary and Singular Metrics: Spaces of psc metrics with boundary convexity/minimality conditions are often contractible or path-connected (Carlotto et al., 2021, Carlotto et al., 2019). Miao's smoothing procedures and Ricci flow extend such results to manifolds with corners or skeleton singularities (Li et al., 2017).

A summary table of moduli phenomena:

Situation	Topology of Moduli Space	Source
High-dim. simply-connected spin, $n\ge5$	Infinitely many components, rich higher homotopy	(Hanke et al., 2012, Crowley et al., 2012, Wiemeler, 2016)
Manifolds with symmetry ($S^1$,$T^n$ actions)	Contractible	(Baer et al., 20 Mar 2025)
3-manifolds with boundary, mean-convex/minimal	Path-connected (sometimes contractible)	(Carlotto et al., 2019, Carlotto et al., 2021)
Boundary moduli under K-area/infinite genus	Nontrivial higher homotopy	(Baer et al., 2020)
Totally nonspin with spin boundary	Nonconnected, nontrivial $\pi_1$	(Frenck, 2022)

5. Nonspin and Singular Phenomena, Generalizations

Studies of psc metric spaces on nonspin manifolds with spin boundary show new rigidity and nonpropagation features:

• Totally Nonspin Manifolds: For oriented manifolds with nonspin universal cover and spin boundary, spaces of psc metrics may have nontrivial fundamental group and are sometimes disconnected, even when natural spin-boundary detection techniques fail (Frenck, 2022).
• Singular Metrics: Singular $L^\infty$ metrics with codimension-2 edge angles $\leq2\pi$ or skeleton singularities do not alter the Yamabe invariant or support "extra" psc metrics unless classical smooth obstructions vanish; smoothing procedures confirm this rigidity (Li et al., 2017).
• Boundary Mean Curvature Smoothing: Deformation principles for boundary mean curvature imply weak homotopy equivalences between spaces with $H\ge0$, $H=0$, or totally geodesic boundary conditions, provided suitable analytic boundary control (Baer et al., 2020).

6. Index Invariants, Group Theory, and Quantitative Lower Bounds

The structure of concordance and moduli groups of psc metrics on $M$ is often sharply bounded by analytic and group-theoretic invariants:

• Finite Part $K$-theory: If $\pi_1(M)$ has torsion, the rank of the moduli group $P(M)$ is bounded below by the number of distinct finite orders in $\pi_1(M)$; presence of torsion forces infinitely many components in the moduli space (Xie et al., 2013).
• Corank of Classifying Map: For closed spin $M$ with $\dim M\ge5$, the rank of the group of psc metrics up to bordism is bounded below by the corank of the KO-homology classifying map $KO_*(M)\to KO_*(B\pi_1(M))$, assuming the rational analytic Novikov conjecture (Schick et al., 2020).
• Higher Rho Invariants: Moduli phenomena are detected by secondary index invariants in localization algebras and $K$-theory (Xie et al., 2013).
• Infinite Rank Lattices: In favorable homological situations, these mechanisms produce moduli groups of infinite rank, even for torsion-free $\pi_1(M)$ (Schick et al., 2020).

7. Open Problems and Future Directions

Major areas of ongoing work include:

• Complete determination of moduli spaces $\Riem^+(M)/\Diff(M)$ for broad classes of manifolds, especially nonspin and with singularities.
• Geometric/topological realization of infinite-order classes in higher homotopy of moduli spaces, notably whether such classes survive full diffeomorphism quotient in arbitrarily high dimension (Hanke et al., 2012, Crowley et al., 2012, Wiemeler, 2016).
• Torsion-free group case: detect new nontrivial psc phenomena not accounted for by finite part K-theory or classical index obstructions (Schick et al., 2020).
• Extension to singular, polyhedral, and skeleton metrics; understanding intrinsic-flat limits and analogs of Ricci flow in singular settings (Li et al., 2017).
• Exploration of boundary effects, especially for psc metrics satisfying mean curvature or second fundamental form constraints, and the impact on the global moduli topology (Baer et al., 2020, Carlotto et al., 2019).

In summary, the theory of positive scalar curvature metrics intertwines analysis, topology, and group theory via rich interactions of index invariants, surgery, boundary phenomena, and moduli topology. The subject remains active, with deep connections to high-dimensional manifold topology, foliation theory, and the foundational structure of geometric analysis.