Gromov's Conjecture in Geometry

• Gromov's Conjecture is a collection of influential problems predicting rigidity and extremality in Riemannian geometry and metric topology.
• It applies analytic, index-theoretic, and surgery techniques to establish scalar curvature bounds and dihedral rigidity in manifolds and polyhedral domains.
• The conjectures connect curvature constraints with topological invariants and group-theoretic properties, driving ongoing advances in geometric analysis.

Gromov’s Conjecture encompasses a suite of influential problems and predictions in global Riemannian geometry, metric topology, group theory, curvature extremality, and index-theoretic rigidity. These conjectures address scalar curvature bounds, geometric collapse, group-theoretic structure, and the relationships between curvature constraints and topological invariants across a wide spectrum of contexts. Many of these statements have motivated extensive research, led to major breakthroughs, or have become central benchmarks in comparison geometry and geometric analysis.

1. Scalar Curvature Extremality and Flat-Domination Conjectures

Gromov formulated geometric maximality principles for domains under scalar curvature and boundary mean curvature constraints. The “flat-corner domination” conjecture asserts that for any convex polyhedron $P \subset \mathbb{R}^n$, if a smooth metric $g$ satisfies

• $\mathrm{Sc}(g) \ge 0$ in $\operatorname{int} P$,
• $H_g(F_i) \ge 0$ for each codimension-1 face $F_i$,
• $\theta_{ij}(g) \le \theta_{ij}(g_0)$ for every pairing of adjacent faces, then $g$ is necessarily flat, all faces are flat, and all ambient angles coincide with those of the standard flat metric $g_0$ (Wang et al., 2022). No strictly "larger" Riemannian metric with nonnegative scalar and mean curvature can be fitted into $P^n$ without reproducing the Euclidean structure, including the corner angle data. This principle extends in rigidity form: strictly convex smooth domains exhibit the same extremality—any metric with nonnegative scalar curvature and nondecreasing boundary convexity is globally rigid.

The analytic and index-theoretic approach, utilizing twisted Dirac operators and parallel spinors, underpins proofs of rigidity and extremality. The existence of sufficiently many parallel sections, induced by Clifford action with respect to flat ambient normals, annihilates all curvature components of the metric, forcing flatness (Wang et al., 2022). These methods generalize Atiyah–Patodi–Singer index theory to polyhedral domains, provided the dihedral angles are all less than $\pi$.

2. Dihedral Rigidity and Extremality in Polyhedra

Gromov’s dihedral rigidity and extremality conjectures posit that, among all metrics on a polyhedron whose scalar curvature is nonnegative and whose dihedral angles are no larger than the Euclidean angles, the Euclidean metric is extremal for invariants such as total volume and boundary measures (Wang et al., 2021). The proofs leverage new index theory for manifolds with polyhedral boundary, incorporating local Clifford boundary conditions and nonnegativity of Dirac operator indices. Rigidity holds under matching dihedral angles: the only metric with nonnegative scalar curvature, mean convex faces, and matching angles is the flat metric.

In the case of acute angles, smoothing constructions and delicate Gauss map interpolations make possible a detailed analytic proof (Brendle et al., 2023). By regulating the Morrey norm of the “error” in curvature mismatch, the boundary integrals in the index computation can be rendered negligible. A parallel spinor then forces flatness and total geodesicity of the faces, yielding complete rigidity.

However, a counterexample exposes the necessity of extra hypotheses regarding index computations on manifolds with corners. The failure of standard APS gluing at vertices requires either stricter angle conditions, smoothing, or sophisticated microlocal analysis (Baer et al., 2022).

3. Macroscopic Dimension and Positive Scalar Curvature

Gromov’s macroscopic-dimension conjecture links curvature positivity and large-scale geometry: if a closed $n$-manifold admits a metric with positive scalar curvature, its universal cover should admit a proper, uniformly cobounded map to an $(n-2)$-dimensional complex, i.e., $\dim_{mc} \tilde{M} \leq n-2$ (Howladar, 31 Mar 2025, Dranishnikov et al., 2023, Bolotov et al., 2014, Dranishnikov, 2013). This prediction, supported for spin, almost-spin, and totally non-spin manifolds with suitable fundamental groups (duality, Baumslag–Solitar, right-angled Artin, etc.), relies on high-level surgery theory, controlled bordism, index-theoretic vanishing, and explicit group cohomological computations. Inessentiality of the classifying map and vanishing of the Dirac index are central technical ingredients. For totally non-spin manifolds, the challenge is topological—index obstructions are unavailable, so an elaborate controlled-surgery argument replaces analytic techniques (Bolotov et al., 2014).

Key group-theoretic reductions demonstrate that, for many fundamental groups (e.g., products of Baumslag–Solitar and one-relator groups), virtual 2-avoidability suffices to establish the conjecture via group homology and Künneth theorems (Howladar, 31 Mar 2025). For right-angled Artin groups, stable splitting of Salvetti complexes provides direct verification via connective $KO$-theory (Dranishnikov et al., 2023).

4. Scalar Curvature Comparison Inequalities (Cube, Bandwidth, Width, Lipschitz Rigidity)

Specific comparison inequalities originated by Gromov—cube inequality, band-width inequality, width estimate for overtorical bands—set optimal geometric bounds on domains with curvature lower bounds. The cube inequality states: for an $n$-dimensional cube $I^n$ with a positive scalar curvature metric,

$$\sum_{i=1}^n \frac{1}{\ell_i^2} \geq \frac{k n}{4 \pi^2 (n-1)},$$

where $\ell_i$ are the $g$-distances between opposite faces, and $k$ is the scalar curvature lower bound (Wang et al., 2021). Strict inequalities, and optimal constants, are achieved using Dirac operator methods—Callias-type operators, Bott-index, and spectral gap arguments replace minimal-surface techniques, which fail in high dimension due to singularities.

The band-width conjecture and associated results for overtorical bands and hemispheres, often proved using relative index theory, establish sharp bounds on width and Lipschitz constants under scalar curvature constraints (Xie, 2021, Zhu, 2020). Rigidity emerges in equality cases—universal covers must be doubly-warped products, and explicit characterization of metrics is possible.

5. Fundamental Group Structure and Ricci Lower Bounds

A prominent conjecture resolves Margulis-type group-theoretic restrictions for manifolds with Ricci curvature bounds. For any complete $n$-manifold $\mathrm{Ric} \geq -(n-1)K$, loops of sufficiently small length generate a subgroup containing a nilpotent subgroup of rank $\leq n$ and index $\leq C(n, K)$ (Kapovitch et al., 2011). The proof deploys rescaling limits, Gromov–Hausdorff convergence, harmonics, and equivariant collapse theory, yielding finiteness results for possible finite group quotients modulo nilpotent cores.

In the setting of positive isotropic curvature, Gromov conjectured that macroscopic 1-dimensionality at scales above the curvature threshold forces the fundamental group to be virtually free, a claim established via Donaldson–Hörmander techniques and filling-radius bounds (Nave, 2013).

6. Euler Characteristic and Signature Inequalities

The Gromov–Lück conjecture asserts that for closed, aspherical $4$-manifolds,

$\chi(X) \geq |\sigma(X)|,$

i.e., the Euler characteristic dominates the absolute value of the signature (Albanese et al., 2023). For complex surfaces, refined classification and $L^2$-index techniques connect this to the Singer conjecture, yielding even sharper bounds (e.g., $\chi \geq \frac{9}{5}|\sigma|$ except possibly in tightly constrained classes). These relationships integrate deep algebraic geometry, $L^2$-Betti number stabilization, and the geography of complex surfaces.

7. Failure of Bounded Cohomology Conjectures

A specific Gromov conjecture predicted equivalence between bounded primitives of differential forms on universal covers and bounded cohomology classes. A counterexample demonstrates that there exist finitely presented groups with weakly bounded but not bounded second cohomology classes, refuting the conjecture (Ascari et al., 2022). The construction relies on amalgamated product presentations, linear isoperimetric inequalities, and delicate analysis of bar complex cochains.

8. Isometric Immersion Theory and Metric-Inducing Operators

Extending beyond geometric analysis, Gromov conjectured that the infinitesimal invertibility (“openness”) of the metric-inducing map for non-free isometric immersions persists well below Nash’s classical dimension threshold. This was affirmed: even for target dimension $q > s_n$ ($s_n = n(n+1)/2$), a dense open set of immersions into $\mathbb{R}^q$ admits local metric realization (1711.01709). The proof synthesizes linearization, upper totally symmetric PDO theory, and the general h-principle.

Table: Forms of Gromov’s Conjecture

Conjecture Type	Statement Prototype	Main Technical Principle
Flat-Corner Domination	“Extremality of Euclidean polyhedra for curvature data”	Dirac index theory, parallel spinors
Dihedral Rigidity/Extremality	“Euclidean metric minimizes volume under angle constraints”	Polyhedral index, smoothing
Macroscopic Dimension PSC	$\dim_{mc} \tilde{M} \leq n-2$ for PSC manifolds	Surgery/obstruction theory, index
Cube/Band-Width Inequalities	Bounds on widths for positive curvature domains	Dirac operators, spectral gap
Margulis Group Structure	Finiteness/rank for small loops under Ricci lower bound	Collapse/rescaling, harmonic functions
Euler Characteristic/Signature	$\chi(X) \geq |\sigma(X)|$ for aspherical 4-manifolds	$L^2$-index, complex surface classification
Bounded Cohomology Equivalence	$\widetilde{d}$-bounded forms $\iff$ bounded cohomology class	Amalgamated group, isoperimetric counterexamples
Metric-Inducing Openness	Infinitesimal invertibility below Nash dimension cutoff	Upper symmetric PDO, h-principle

Concluding Remarks

Gromov’s Conjecture(s) establish a landscape of sharp geometric rigidity phenomena and group-theoretic constraints at the intersection of curvature, topology, and large-scale geometry. The impact of these conjectures is manifest in the development of Dirac-index methods for scalar curvature, advances in surgery techniques, new rigidity results for polyhedral and band domains, and group-theoretic classification under geometric metric conditions. A persistent theme is the intimate connection between analytic invariants (indices, curvature, spectral gaps) and topological features (classifying map, skeleton dimension, group presentations), with deep implications for both differential geometry and algebraic topology. Several open questions remain, especially regarding generalization to arbitrary fundamental groups, extension to non-acute angle cases, and further classification results in four-manifold geometry.