Entropy Collapse in 4-Manifolds

• Entropy collapse is a phenomenon where geometric invariants like minimal entropy and volume vanish in symplectic 4-manifolds due to F-structures.
• It is achieved by constructing sequences of metrics with sectional curvature bounded from below, linking geodesic flow complexity with curvature behavior.
• This collapse unifies aspects of topology, symplectic geometry, and scalar curvature, revealing rigidity in the Yamabe invariant across Kodaira dimensions.

Entropy collapse refers to the phenomenon in which geometric, dynamical, or statistical invariants that measure complexity, diversity, or disorder—such as topological entropy, minimal volume, or scalar curvature invariants—vanish or are drastically reduced in certain settings. In the context of symplectic and Riemannian geometry, entropy collapse describes scenarios where the minimal entropy, minimal volume, and related invariants of symplectic 4-manifolds fall to zero as a result of geometric collapse endowed with specific structures (notably F-structures or T-structures). This collapse reveals profound relationships between geometric topology, symplectic structures, curvature, and the scalar curvature spectrum as encapsulated by the Yamabe invariant. The phenomenon has far-reaching consequences for the paper of geometric invariants, curvature collapse, and the topological flexibility of symplectic and Riemannian 4-manifolds.

1. Minimal Entropy and Minimal Volume: Definitions and Computation

Minimal entropy, denoted $h(M)$ for a closed oriented manifold $M$, is defined as the infimum of topological entropy of the geodesic flow over all smooth metrics: $$h(M) = \inf\{ h_{\text{top}}(g) : g \text{ is a smooth metric on } M \}.$$ Minimal volume is the infimum of manifold volumes among all smooth metrics with sectional curvature constrained to lie in $[-1,1]$ (or another prescribed lower or two-sided bound): $$\text{Vol}_K(M) = \inf \{ \text{Vol}(M,g) : K(g) \geq -1 \},$$ often with normalization $\text{Vol}(M,g)=1$ to facilitate comparisons.

Another related invariant, Volk$(M)$, takes the infimum of volumes over all metrics with curvature bounded below. All these invariants measure the "geometric complexity" and behavior of the manifold under scaling and curvature constraints.

A central technique in establishing entropy collapse is the construction of F-structures (and, as a special case, T-structures). The presence of an F-structure (a geometric structure built from torus actions possibly with singularities and gluing) provides control over the collapse process. Cheeger–Gromov, Paternain–Petean, and subsequent authors proved that once such a structure is present, it forces

$$h(M) = 0, \quad \|M\| = 0, \quad \text{Vol}_K(M) = 0,$$

signifying that the manifold can be collapsed (with controlled curvature) to arbitrarily small volume while its entropy, a key invariant of geodesic flow complexity, drops to zero. In this context, "entropy collapse" refers to the simultaneous vanishing of these key invariants under the geometric process enabled by the F-structure.

2. Collapse with Sectional Curvature Bounded from Below

The collapse mechanism, central to entropy collapse, is characterized by the existence of sequences of metrics with sectional curvature uniformly bounded from below (i.e., $K \geq -1$) and with normalizations such that the volume tends to zero. The main result formalizes: $$\text{If } M \text{ admits an F-structure, then} \quad h(M) = 0, \quad \|M\| = 0, \quad \text{Vol}_{\text{Ric}}(M) = 0,$$ and the manifold collapses with curvature $K \geq -1$.

F-structures are constructed by gluing together pieces of the manifold, each equipped with torus actions, and identifying along boundaries with affine maps. These structures are robust in the symplectic field, often achievable by careful surgeries or symplectic sums. The existence of an F-structure thus suffices to trigger entropy collapse for a wide class of symplectic 4-manifolds.

3. Vanishing of the Yamabe Invariant and its Relation to Collapse

The Yamabe invariant $y(M)$ of a compact manifold $M$ is defined as the supremum over conformal classes $[g]$ of the infimum of total scalar curvature normalized by volume (raised to the $(2/n)$ power): $$y(M) = \sup_{[g]} \inf_{g \in [g]} \frac{\int_M \operatorname{Scal}_g \, \operatorname{dvol}_g}{(\operatorname{Vol}(M,g))^{2/n}}.$$ On symplectic 4-manifolds with F-structures and non-vanishing Seiberg–Witten invariants (typical when the Kodaira dimension satisfies $\mathrm{kod}(M,\omega) \leq 1$), the minimal entropy and minimal volume vanish, and crucially, the Yamabe invariant also vanishes: $y(M) = 0.$ However, such manifolds often cannot support metrics of positive scalar curvature or even scalar-flat metrics realizing $y(M)=0$, especially in Kodaira dimensions zero and one—a rigidity consequence relying on the classification of Ricci-flat and scalar-flat metrics by Hitchin.

This vanishing illustrates the deep link between scalar curvature geometry and entropy collapse: geometric flattening through F-structure collapse drives both entropy and Yamabe invariants to their minimizers, but in highly constrained metric spaces.

4. Extension of LeBrun’s Kodaira Dimension–Yamabe Link to the Symplectic Category

LeBrun’s result originally established a connection between the holomorphic Kodaira dimension and the Yamabe invariant for compact complex surfaces. The paper extends this relationship to symplectic 4-manifolds endowed with symplectic Kodaira dimension $\mathrm{kod}(M,\omega)$ (per Li's symplectic classification).

Key results established:

• $y(M)>0$ if $\mathrm{kod}(M,\omega) = -\infty$
• $y(M)=0$ if $\mathrm{kod}(M,\omega) = 0$ or $1$
• $y(M)<0$ if $\mathrm{kod}(M,\omega) = 2$

This extension reveals that entropy collapse (disappearance of minimal volume, entropy, and related invariants) persists across the Kodaira dimension hierarchy, provided an F-structure is present. Particularly, for Kodaira dimension one, the zero Yamabe invariant is not realized by any metric, emphasizing the subtlety and rigidity of the geometry even as collapse occurs.

5. Construction Flexibility: Realization of Arbitrary Fundamental Groups

Symplectic 4-manifolds show extraordinary topological flexibility: any finitely presented group $G$ can be realized as the fundamental group $\pi_1(M)$ of a symplectic 4-manifold $M$. By Gompf’s construction (and refinements by Baldridge–Kirk), one can build $M(G)$ such that

$$h(M(G)) = 0, \quad \|M(G)\| = 0, \quad \text{Vol}_K(M(G)) = 0, \quad y(M(G))=0$$

and that $M(G)$ collapses with sectional curvature bounded from below.

This universality shows that entropy collapse is not a rare or pathological event but generic among symplectic 4-manifolds, regardless of topological complexity.

6. Synthesis: Interplay of Topology, Symplectic Geometry, and Scalar Curvature

The analysis reveals profound interplay:

• Entropy collapse reflects a type of geometric flattening, whereby the complexity of geodesic flow and the metric volume can be driven to arbitrarily small values via F-structure collapse.
• The vanishing or non-attainment of the Yamabe invariant connects analytic (scalar curvature) and geometric-topological properties.
• Extension of LeBrun’s Kodaira dimension–Yamabe result to symplectic 4-manifolds situates entropy collapse within the broad classification paradigms of both complex and symplectic geometry.

The phenomenon also highlights the differences between topological and curvature constraints: while adding more topological complexity (e.g., arbitrary fundamental groups) is fully compatible with entropy collapse, the rigidity of scalar curvature and Seiberg–Witten theory imparts strong restrictions on which metrics realize the limiting values.

Table: Collapse Phenomena in Symplectic 4-Manifolds

Invariant	Collapse Under F-structure	Comments
Minimal Entropy $h$	$h(M) = 0$	Geodesic flow complexity vanishes
Minimal Volume	$\text{Vol}_K(M) = 0$	With $K \geq -1$ curvature bound
Yamabe Invariant $y$	$y(M)=0$ (Kod$_{\leq 1}$)	Not realized in metric for Kod=1
Fundamental Group	Arbitrary $G$ allowed	Flexibility does not obstruct collapse

Conclusion and Significance

Entropy collapse in symplectic 4-manifolds illustrates how geometric invariants can vanish in the presence of special topological and symplectic structures, particularly F-structures, signaling extreme flexibility of 4-manifolds under appropriate geometric constraints. The phenomenon underscores the intricate connections between topology, F-structure collapse, scalar curvature, and the analytic landscape of the Yamabe problem. Advancements in linking Kodaira dimension and collapse phenomena further unify themes in complex, symplectic, and Riemannian geometry and indicate rich terrain for future exploration in the geometry of high-dimensional manifolds and the analytic structures underlying geometric collapse (Suárez-Serrato et al., 2014).