Optimal Systolic Inequalities

• Optimal systolic inequality is a sharp bound linking shortest noncontractible loops (systoles) with volumetric measures to classify extremal metrics.
• It employs techniques like cutting-and-pasting, calibration, and symplectic methods to establish precise constants and quantify deviations from optimality.
• Applications span Riemannian, Finsler, contact, and polyhedral geometries, illuminating interplays between curvature, topology, and global analysis.

An optimal systolic inequality is a sharp bound relating a geometric invariant—typically the length of the shortest non-contractible or homologically non-trivial loop or cycle, called the systole—to a volumetric invariant such as area, volume, or contact volume, on a closed Riemannian or Finsler manifold, or more generally for contact or polyhedral structures. The inequality is "optimal" when its constant is sharp and, if possible, the extremal metrics or structures are classified. Optimal systolic inequalities underlie major advances in differential geometry, global analysis, metric geometry, and topology and exhibit subtle interplays between curvature, homology, combinatorics, and symplectic dynamics.

1. Systolic Invariants and Systolic Ratios

Let $(M,g)$ be a closed, oriented Riemannian $m$-manifold. The systole $\sys(M,g)$ is defined as the length of the shortest non-contractible closed loop in $M$, i.e.,

$\sys(M,g) := \inf \big\{ \ell_g(\gamma)\ |\ \gamma \subset M\ \text{closed, non-contractible} \big\}.$

The normalized systolic ratio is then

$SR(M,g) := \frac{\sys(M,g)^m}{\Vol(M,g)},$

which is invariant under rescaling of the metric. For surfaces, the supremum of the systolic ratio in genus $k$,

$$SR(k) := \sup_{(S,g)\ \text{closed oriented of genus }k}\ SR(S,g),$$

defines the optimal $k$-genus systolic constant, whose reciprocal, $\sigma(k) := \inf \Vol(S,g)/\sys(S,g)^2$, is called the optimal systolic area in genus $k$ (Akrout et al., 2013).

In addition, one distinguishes homological systoles, where $\sys_h(S,g)$ is the length of the shortest homologically non-trivial closed loop, and $SR_h(S,g) := \sys_h(S,g)^2/\Vol(S,g)$, with $SR_h(k) := \sup SR_h(S,g)$ in genus $k$.

The notion of systolic ratio extends to higher homology (e.g., the $p$-homology systole) and to contact, Finsler, and polyhedral categories, as well as to other settings like 2-complexes and group cohomologies (Borghini, 2019).

2. Canonical and Extremal Cases: Classical Results

The prototype for optimal systolic inequalities is Loewner's torus inequality,

$\Area(T^2,g) \ge \frac{\sqrt{3}}{2} \sys(T^2,g)^2,$

with equality for the flat torus with hexagonal lattice. For the real projective plane, Pu's theorem (Katz et al., 2020) states

$\sys(\RP^2,g)^2 \le \frac{\pi}{2}\Area(\RP^2,g),$

with equality for the round metric. Pu's inequality admits Bonnesen-type refinements with remainder terms quantifying deviation from roundness.

On the Klein bottle, Bavard's theorem establishes $\alpha_{sys}(K,g) := \area(K,g)/\sys^2(K,g) \ge 2\sqrt{2}/\pi$, achieving equality in a specific (non-smooth) conformal class (Eyll, 19 Feb 2025). For the Möbius band and more general surfaces, analogous sharp results control the area (or Holmes-Thompson/Finsler volume) in terms of the product of systole and geometric width or height (Sabourau et al., 2015).

On the 2-sphere, an optimal inequality under positive curvature pinching asserts

$\ell_{\min}(g)^2 < \pi\Area(S^2,g) \le \ell_{\max}(g)^2,$

where equality holds only for Zoll metrics (i.e., metrics all of whose geodesics are closed of the same length) (Abbondandolo et al., 2014).

3. Optimal Systolic Inequalities for Higher Genus and Asymptotics

The determination of $SR(k)$ for surfaces of genus $k$ is a fundamental problem. The intersystolic (or "summation") inequality (Akrout et al., 2013)

$$\frac{1}{SR(k_1 + k_2)} \leq \frac{1}{SR(k_1)} + \frac{1}{SR(k_2)}$$

is established by careful cutting and pasting of nearly extremal surfaces: gluing halves of surfaces with high systolic ratio along boundary geodesics preserves the systole and yields controllable volume in the resulting higher-genus surface. The same construction holds for the homological systolic ratio $SR_h$.

Sharp asymptotics for large genus are known: $$\frac{4}{9\pi}\frac{(\log k)^2}{k} \lesssim SR(k) \lesssim \frac{1}{\pi} \frac{(\log k)^2}{k},$$ where arithmetic/hyperbolic examples (Buser–Sarnak, Katz–Schaps–Vishne) yield the lower bound and Gromov's techniques push down the upper bound for $k \ge 76$ (for $SR_h$) (Akrout et al., 2013). The gap between the two constants remains open; it is conjectured that both $SR(k)$ and $SR_h(k)$ tend to $c(\log k)^2/k$ with the same constant $c = 1/\pi$.

For $k=1$, Loewner's theorem is sharp: $SR(1) = 2/\sqrt{3} \approx 1.1547$.

4. Systolic Inequalities in Higher Dimensions and Group/Complex Settings

In dimension $n \geq 3$, Gromov generalized systolic inequalities using the cup-length and combinatorial complexity of the underlying space (Chen, 2015, Borghini, 2019). For closed aspherical 3-manifolds $M$ of triangulation complexity $c(M)$, the systolic volume satisfies

$\sigma(M) \geq C\exp(C'\sqrt{\ln c(M)}),$

providing a substantially stronger lower bound than those depending on Betti numbers or simplicial volume.

For piecewise Riemannian 2-complexes $X$ where $H^1(X;\Z_2)$ admits two classes with nonzero cup product, Borghini established the sharp relation

$\sys(X,g)^2 \leq \Vol(X,g)$

(Borghini, 2019), extending earlier results of Guth to the singular (complex) setting. The systolic area of groups containing "surface-like" subgroups satisfies the optimal universal lower bound $\sigma(G)\ge 2$.

For closed Riemannian $(n+1)$-manifolds with nontrivial $H_n$ and positive bi-Ricci curvature ($\biRic_g > n-1$), Chu–Lee–Zhu proved

$\sys_n^H(M,g) \le |S^n|,$

with rigidity: equality occurs only when the universal cover splits isometrically as $S^n \times \R$ (Chu et al., 2024). This generalizes the Toponogov–Bray–Brendle–Neves rigidity phenomena and, in codimension-2, the analogous statement holds under positive triRic curvature (Chen et al., 18 Jan 2026).

5. Methods of Proof: Cut-and-Paste, Calibration, Symplectic and Contact Structures

The proofs of optimal systolic inequalities use a variety of techniques:

• Cutting and pasting: Decomposition of extremal or nearly extremal metrics into pieces, glued along geodesics or along minimal surfaces, preserves systole under carefully controlled homotopy and volume considerations (Akrout et al., 2013).
• Calibration: The optimal constants in cohomological systolic inequalities are frequently realized via calibration forms, notably in settings with cup-product decomposable fundamental class or stable norm representations (Goodwillie et al., 2024).
• Symplectic/contact geometry: For Riemannian and Finsler manifolds whose geodesic flows can be recast as Reeb flows of contact manifolds, invariants such as the Calabi invariant, generating functions, and fixed-point theorems in symplectic dynamics play a key role. In the Zoll case, the systolic ratio is locally maximized among contact forms and Finsler metrics near the standard structure, with full rigidity of equality cases (Abbondandolo et al., 2019, Abbondandolo et al., 2014).
• Stability and remainder terms: Bonnesen-style refinements quantify how deviation from extremal metrics is controlled by conformal or $L^2$-distance from the extremal factor, with precise lower bounds for the systolic defect (Katz et al., 2020, Eyll, 19 Feb 2025).
• Dimensional reduction/slicing: For higher-dimension or codimension, inductive slicing arguments reduce the problem to two-dimensional inequalities for minimal surfaces, with the curvature assumptions descending under the slice (Brendle et al., 2024, Chu et al., 2024, Chen et al., 18 Jan 2026).

6. Systolic Inequalities in Contact Geometry and Finsler Metrics

Beyond Riemannian settings, optimal systolic inequalities extend naturally to contact geometry. For 3-dimensional Seifert bundles with nonzero Euler number, all $S^1$-invariant contact forms $\alpha$ satisfy

$\sys(\alpha)^2 \le C \max(1,1/|e|)\,\Vol(\alpha)$

with optimality and equality for Zoll contact forms arising from the Boothby–Wang construction (Vialaret, 2024). Stability and sharpness are demonstrated in the transverse case, while the optimal constant in general remains unresolved.

Finsler and Holmes–Thompson volumes play a central role in systolic theory for non-Riemannian metrics, with dimension-2 extremals occurring for parallelogram norms or more intricate systolic-calibrated bodies (Balacheff et al., 2022, Sabourau et al., 2015). The classification of extremals may demand consideration of polyhedral or non-smooth structures, and the absence of uniformization severely limits Riemannian-specific tools.

7. Open Problems and Future Directions

Several questions remain at the frontier:

• Determining the precise asymptotic constant in $SR(k) \sim c(\log k)^2/k$ for large genus surfaces, and the validity of the conjecture $SR(k) = SR_h(k)$ with $c=1/\pi$ (Akrout et al., 2013).
• Characterizing monotonicity properties such as $SR(k+1)\le SR(k)$, which is unresolved.
• Optimizing the explicit constants in contact systolic inequalities, especially for non-transverse forms or in higher dimensions (Vialaret, 2024).
• Extending systolic inequalities to singular spaces, polyhedral complexes, or groups beyond the "surface-like" regime (Chen, 2015, Borghini, 2019).
• Developing sharp stability and remainder theorems quantifying distance to extremality in a global geometric or conformal sense (Eyll, 19 Feb 2025, Katz et al., 2020).
• Exploring implications in mathematical physics, particularly through connections with positive energy theorems and the rigidity of asymptotically locally hyperbolic spaces (Brendle et al., 2024).

Optimal systolic inequalities serve as a nexus of geometric analysis, metric topology, and algebraic topology, embodying the depth and subtlety of global differential geometry across manifold and complex settings.