Metric-Curvature Inequalities

• Metric-curvature inequalities quantitatively link scalar curvature lower bounds to metric, transport, and spectral properties in Riemannian geometry.
• The Dirac-operator proof extends Gromov’s cube inequality to all dimensions, establishing optimal constants and overcoming previous dimensional limits.
• Synthetic formulations connect these inequalities to Bakry–Émery conditions, optimal transport, and isoperimetric estimates, broadening their application across geometric analysis.

Searching arXiv for recent and foundational papers directly relevant to metric–curvature inequalities. arXiv search: "metric curvature inequality scalar curvature cube inequality" A metric–curvature inequality is an estimate in which a curvature bound controls a metric, measure, transport, spectral, or bundle-theoretic quantity. In the narrow sense used by Wang–Xie–Yu, it denotes the sharp scalar-curvature estimate for a Riemannian cube: if $(I^n,g)$ has $\mathrm{Scal}_g\ge k>0$ and $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$ is the distance between opposite faces, then

$$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$

This yields the optimal constant $4\pi^2$ in all dimensions and strengthens Gromov’s earlier minimal-surface result, which was known only for $n\le 8$ (Wang et al., 2021). The literature also uses closely related inequalities in synthetic Ricci geometry, isoperimetry, Finsler and weighted settings, and operator theory; this suggests a general paradigm in which curvature bounds constrain the geometry of distance, volume, entropy, and localization.

1. Scalar-curvature origins

A central source of the subject is Gromov’s program of deriving metric inequalities from scalar-curvature lower bounds. In "Metric Inequalities with Scalar Curvature" Gromov establishes estimates relating $\mathrm{Sc}(g)\ge \kappa>0$ to distances, widths of bands, Lipschitz constants of maps, diameters of focal tubes, and depths of homology classes (Gromov, 2017).

For a band $V$ with distinguished boundary components $\partial_-V$ and $\partial_+V$, the basic metric quantity is the width

$\mathrm{Scal}_g\ge k>0$0

If $\mathrm{Scal}_g\ge k>0$1 is a torical band and $\mathrm{Scal}_g\ge k>0$2, Gromov proves the torical $\mathrm{Scal}_g\ge k>0$3-inequality

$\mathrm{Scal}_g\ge k>0$4

Under the normalization $\mathrm{Scal}_g\ge k>0$5, this becomes $\mathrm{Scal}_g\ge k>0$6 (Gromov, 2017).

The same paper gives $\mathrm{Scal}_g\ge k>0$7-bounds for wider topological classes. If $\mathrm{Scal}_g\ge k>0$8 is an iso-enlargeable band or a SYS-band with $\mathrm{Scal}_g\ge k>0$9, then

$\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$0

For complete SYSE-manifolds the corresponding SYSE-width satisfies the same upper bound (Gromov, 2017). Gromov also proves a sub-rectangular inequality: if $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$1 satisfies $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$2 and $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$3 is diffeomorphic to $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$4 with all faces except the top and bottom mean-convex and all dihedral angles $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$5, then

$\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$6

These estimates place the cube inequality in a broader scalar-curvature framework in which lower scalar curvature acts as an obstruction to large widths and large-distance configurations (Gromov, 2017).

2. Gromov’s cube inequality and its Dirac-operator proof

For the standard cube $\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$7, let

$\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$8

and define

$\ell_i=\mathrm{dist}_g(d_i^{-},d_i^{+})$9

Wang–Xie–Yu prove that if $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$0 on $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$1, then

$$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$2

In particular, when $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$3,

$$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$4

If in addition all dihedral angles of $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$5 are $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$6, then the inequality improves to a strict inequality (Wang et al., 2021).

The result resolves the dimensional restriction in Gromov’s original proof. The minimal-surface argument worked only for $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$7, whereas the Dirac-operator method gives the optimal constant in all dimensions. The paper also states that a family of warped-product examples shows that $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$8 is best possible (Wang et al., 2021).

The analytic mechanism is a Callias-type deformation of a Dirac operator on $$\sum_{i=1}^n \frac1{\ell_i^2}\;\ge\;\frac{n\,k}{4\pi^2\,(n-1)}.$$9. One first extends $4\pi^2$0 from $4\pi^2$1 to a complete metric on $4\pi^2$2 that is Euclidean outside a compact set, while preserving $4\pi^2$3 on $4\pi^2$4 after scaling and arranging $4\pi^2$5 on $4\pi^2$6. For each coordinate direction one introduces signed-distance functions $4\pi^2$7 and approximate slab functions $4\pi^2$8 with

$4\pi^2$9

On the spinor bundle

$n\le 8$0

with $n\le 8$1, one considers the twisted Dirac operator $n\le 8$2, which satisfies the Weitzenböck formula

$n\le 8$3

A potential

$n\le 8$4

is chosen so that each $n\le 8$5 solves

$n\le 8$6

hence $n\le 8$7 on the relevant interval. The Callias operator

$n\le 8$8

then satisfies

$n\le 8$9

Outside a compact set, $\mathrm{Sc}(g)\ge \kappa>0$0, so $\mathrm{Sc}(g)\ge \kappa>0$1 is Fredholm and essentially self-adjoint. A homotopy to the classical Bott–Dirac operator at infinity gives $\mathrm{Sc}(g)\ge \kappa>0$2. On the other hand, a coercivity estimate shows that $\mathrm{Sc}(g)\ge \kappa>0$3 would be invertible if the cube inequality failed. The contradiction between index $\mathrm{Sc}(g)\ge \kappa>0$4 and invertibility forces the metric–curvature bound (Wang et al., 2021).

The strict form for manifolds with corners uses APS-style boundary conditions and corner index theory. In the borderline case, the argument produces a nontrivial kernel which must vanish by unique continuation along the faces, yielding the strict inequality when all dihedral angles are $\mathrm{Sc}(g)\ge \kappa>0$5 (Wang et al., 2021).

3. Synthetic curvature-dimension formulations

In metric measure geometry, metric–curvature inequalities are encoded through transport convexity, Bakry–Émery inequalities, and evolution variational inequalities. Erbar–Kuwada–Sturm define the entropic curvature-dimension condition $\mathrm{Sc}(g)\ge \kappa>0$6 by requiring that for every finite-entropy pair $\mathrm{Sc}(g)\ge \kappa>0$7 there exists a $\mathrm{Sc}(g)\ge \kappa>0$8-geodesic $\mathrm{Sc}(g)\ge \kappa>0$9 such that

$V$0

where $V$1 (Erbar et al., 2013). On infinitesimally Hilbertian spaces this is equivalent to the Bochner inequality $V$2, to the reduced condition $V$3, and to the statement that the heat flow is the $V$4-gradient flow of entropy and satisfies $V$5 (Erbar et al., 2013).

The same equivalence yields explicit transport contraction. In particular,

$V$6

and the space-time estimate

$V$7

holds for all $V$8 (Erbar et al., 2013). This provides a transport-theoretic form of curvature control.

Ketterer extends this picture to variable lower curvature bounds $V$9. The generalized distortion coefficient $\partial_-V$0 is defined as the solution of

$\partial_-V$1

The corresponding $\partial_-V$2 and $\partial_-V$3 conditions recover the constant-curvature theory when $\partial_-V$4; $\partial_-V$5 is stable under Gromov convergence and is equivalent to $\partial_-V$6 on essentially non-branching spaces (Ketterer, 2015). A differential Wasserstein contraction estimate then reads

$\partial_-V$7

in the $\partial_-V$8 regime (Ketterer, 2015).

Metric graphs exhibit a weakened version of the same trinity. Krautz proves that compact metric graphs satisfy a weak Bakry–Émery estimate

$\partial_-V$9

with constants $\partial_+V$0 and $\partial_+V$1, and establishes its equivalence with a weak EVI and a weak form of geodesic convexity of entropy (Krautz, 17 Dec 2025). The paper emphasizes that none of the classical equivalent formulations of $\partial_+V$2 survives intact on such graphs, but all three can be recovered in weak form (Krautz, 17 Dec 2025). This is an explicit example in which the metric–curvature principle persists while exact smooth equivalence fails.

4. Isoperimetric, Heintze–Karcher, and comparison inequalities

A large class of metric–curvature inequalities is isoperimetric. In essentially non-branching $\partial_+V$3 spaces, Cavalletti and Mondino prove a Heintze–Karcher inequality by localizing the signed distance from a boundary $\partial_+V$4 into one-dimensional transport rays (Ketterer, 2019). If $\partial_+V$5 and $\partial_+V$6 is the Jacobian model

$\partial_+V$7

then

$\partial_+V$8

If inner curvature is also defined, one obtains the global estimate

$\partial_+V$9

In $\mathrm{Scal}_g\ge k>0$00 spaces with $\mathrm{Scal}_g\ge k>0$01, equality characterizes spherical suspensions (Ketterer, 2019).

Han proves a sharp dimension-free isoperimetric inequality for non-compact $\mathrm{Scal}_g\ge k>0$02 spaces: $\mathrm{Scal}_g\ge k>0$03 Here $\mathrm{Scal}_g\ge k>0$04 is the volume entropy, and the coefficient is optimal (Han, 2021). The one-dimensional model $\mathrm{Scal}_g\ge k>0$05 realizes equality (Han, 2021).

The nonpositively curved Riemannian setting yields another comparison form. Ghomi–Stavroulakis show that there exists $\mathrm{Scal}_g\ge k>0$06 such that if $\mathrm{Scal}_g\ge k>0$07 is a smooth metric on the Euclidean ball $\mathrm{Scal}_g\ge k>0$08 with $\mathrm{Scal}_g\ge k>0$09 and $\mathrm{Scal}_g\ge k>0$10, then

$\mathrm{Scal}_g\ge k>0$11

with equality if and only if $\mathrm{Scal}_g\ge k>0$12 is isometric to a Euclidean ball, equivalently $\mathrm{Scal}_g\ge k>0$13 up to homothety (Ghomi et al., 17 May 2025). The proof uses Rauch/Bishop–Gromov comparison, a coarea formula in normal coordinates, and a monotone interpolation quantity $\mathrm{Scal}_g\ge k>0$14 (Ghomi et al., 17 May 2025).

Cavalletti–Mondino derive a broad family of sharp inequalities under essentially non-branching $\mathrm{Scal}_g\ge k>0$15: the Brunn–Minkowski inequality

$\mathrm{Scal}_g\ge k>0$16

the sharp $\mathrm{Scal}_g\ge k>0$17-spectral gap $\mathrm{Scal}_g\ge k>0$18, and corresponding sharp log-Sobolev, Talagrand, and Sobolev inequalities (Cavalletti et al., 2015). The proofs proceed through one-dimensional needle decomposition, showing that lower Ricci bounds control measure interpolation along transport rays (Cavalletti et al., 2015).

5. Weighted, Finsler, and analytic rigidity

In smooth metric measure spaces, a metric–curvature inequality can be integral rather than pointwise. Li proves that on a compact weighted manifold $\mathrm{Scal}_g\ge k>0$19 with $\mathrm{Scal}_g\ge k>0$20,

$\mathrm{Scal}_g\ge k>0$21

The constant is exactly the classical De Lellis–Topping constant, and equality forces a weighted Einstein space with constant $\mathrm{Scal}_g\ge k>0$22 (Wu, 2011). This is presented as a mild generalization of the almost-Schur theorem to smooth metric measure spaces (Wu, 2011).

Du–Mao–Wang–Wu show that if a proper metric measure space satisfies a volume doubling condition of exponent $\mathrm{Scal}_g\ge k>0$23 and the Gagliardo–Nirenberg inequality with the same exponent $\mathrm{Scal}_g\ge k>0$24, then it has exactly $\mathrm{Scal}_g\ge k>0$25-dimensional volume growth (Du et al., 2015). In the Finsler setting, if a complete $\mathrm{Scal}_g\ge k>0$26-dimensional Finsler manifold with nonnegative $\mathrm{Scal}_g\ge k>0$27-Ricci curvature satisfies the sharp Euclidean Gagliardo–Nirenberg inequality, then its flag curvature is identically zero (Du et al., 2015). The argument follows the chain

$\mathrm{Scal}_g\ge k>0$28

which the paper explicitly describes as a metric–curvature pattern (Du et al., 2015).

A related Finsler isoperimetric theory appears in the work of Wang and Zhao. For a forward complete non-compact Finsler metric measure manifold with $\mathrm{Scal}_g\ge k>0$29, they define the volume entropy

$\mathrm{Scal}_g\ge k>0$30

and prove the sharp inequality

$\mathrm{Scal}_g\ge k>0$31

They also define the second Cheeger constant $\mathrm{Scal}_g\ge k>0$32 and obtain the Cheeger–Buser type bounds

$\mathrm{Scal}_g\ge k>0$33

When $\mathrm{Scal}_g\ge k>0$34, sharpness gives $\mathrm{Scal}_g\ge k>0$35 (Cheng et al., 11 Jul 2025). This transfers the entropy–isoperimetry relation into the non-reversible Finsler category.

6. Chiral and operator-theoretic extensions

Not all metric–curvature inequalities are formulated in terms of distances or volumes. Fine proves a chiral curvature criterion for four-dimensional Poincaré–Einstein manifolds: if $\mathrm{Scal}_g\ge k>0$36, or symmetrically $\mathrm{Scal}_g\ge k>0$37, then the metric is non-degenerate, meaning that the only $\mathrm{Scal}_g\ge k>0$38-solution of the gauge-fixed linearized Einstein equation is trivial (Fine, 2022). Here $\mathrm{Scal}_g\ge k>0$39 is the self-dual block of the curvature operator $\mathrm{Scal}_g\ge k>0$40, and $\mathrm{Scal}_g\ge k>0$41 means that for every nonzero self-dual $\mathrm{Scal}_g\ge k>0$42-form $\mathrm{Scal}_g\ge k>0$43,

$\mathrm{Scal}_g\ge k>0$44

This strictly generalizes the Biquard–Lee condition of negative sectional curvature in dimension four, because it requires only $\mathrm{Scal}_g\ge k>0$45 or $\mathrm{Scal}_g\ge k>0$46 rather than non-positivity of the full curvature operator (Fine, 2022).

In operator theory, Biswas–Keshari–Misra study curvature inequalities for operators in the Cowen–Douglas class $\mathrm{Scal}_g\ge k>0$47. If $\mathrm{Scal}_g\ge k>0$48 is a contraction, then its bundle curvature satisfies

$\mathrm{Scal}_g\ge k>0$49

However, the converse fails: an operator can satisfy the curvature inequality without being contractive (Biswas et al., 2012). The paper therefore isolates a stronger condition,

$\mathrm{Scal}_g\ge k>0$50

as a positive-definite scalar kernel, and shows that this characterizes infinitely divisible contractions (Biswas et al., 2012). In several variables, the corresponding curvature matrix is compared with that of the Drury–Arveson shift, and infinite divisibility of the kernel becomes equivalent to a positive-definite matrix-valued curvature inequality (Biswas et al., 2012).

These variants clarify a common misconception. A curvature inequality need not have a single universal form, and it need not always compare distances directly. In the material surveyed here, the same phrase covers scalar-curvature bounds on widths of bands and cubes, transport-convexity and contraction inequalities, isoperimetric and Heintze–Karcher comparisons, integral rigidity estimates in weighted geometry, and positivity conditions on curvature kernels in operator theory. The unifying feature is not the formula but the mechanism: curvature bounds impose quantitative restrictions on admissible metric, analytic, or transport behavior (Wang et al., 2021).