Non-Smooth Action-Minimizing Characteristics

• The paper extends classical Zoll domains to non-smooth settings by demonstrating that cuts additivity and generalized Zoll properties are analytically equivalent.
• It details how the Ekeland–Hofer–Zehnder capacity quantifies action-minimizing properties using hyperplane cuts and variational methods.
• The work classifies local dynamical behaviors into extreme rays, coisotropic faces, and isotropic gliding, offering precise criteria for non-smooth periodic dynamics.

Non-smooth action-minimizing closed characteristics arise in the paper of convex bodies in symplectic phase space, particularly as generalizations of classical Zoll properties to domains where smoothness does not necessarily hold. These objects capture the structure of periodic dynamics on the boundaries of convex bodies when classical differentiable tools are no longer fully applicable, extending the interaction between symplectic topology, variational principles, and dynamical systems.

1. Classical Zoll Property and Its Dynamical Extension

In the standard symplectic phase space $$(\mathbb{R}^{2n}, \omega = \sum_i dp_i \wedge dq_i)$$, a smooth, strictly convex, star-shaped domain $K \subset \mathbb{R}^{2n}$ has a boundary $\partial K$ endowed with the canonical contact form

$$\alpha = \frac{1}{2} \sum_{i=1}^{n} (p_i\,dq_i - q_i\,dp_i).$$

The Reeb vector field $R$ associated to $\alpha$ spans the characteristic line bundle $\ker(\omega|_{T\partial K})$. When every Reeb orbit on $\partial K$ is closed and has the same minimal period, $K$ is termed a Zoll domain---the boundary is foliated by action-minimizing closed characteristics of constant period.

In the absence of smoothness, the classical Zoll property admits a dynamical extension termed the “cuts additivity” condition. Given any affine hyperplane $H$ subdividing $K$ into convex $K_1$ and $K_2$, cuts additivity requires that the Ekeland-Hofer-Zehnder capacity satisfies

$$c_{\rm EHZ}(K) = c_{\rm EHZ}(K_1) + c_{\rm EHZ}(K_2).$$

This criterion provides a symmetry-like constraint connecting the global symplectic geometry of $K$ to the behavior of action-minimizing closed characteristics on potentially non-smooth boundaries (Haim-Kislev, 20 Nov 2025).

2. The Ekeland–Hofer–Zehnder Capacity and Hyperplane Cuts

The Ekeland–Hofer–Zehnder (EHZ) capacity $c_{\rm EHZ}(K)$ is a symplectic invariant defined for a convex $K \subset \mathbb{R}^{2n}$ as the minimal action among all closed characteristics $\gamma \subset \partial K$: $$\mathcal{A}(\gamma) = \int_\gamma \alpha, \qquad c_{\rm EHZ}(K) := \min\{\mathcal{A}(\gamma): \gamma \subset \partial K \ \text{is a closed characteristic}\}.$$ This capacity is monotonic under symplectic embeddings and subadditive: whenever $K_1 \subset K_2$, $c_{\rm EHZ}(K_1) \leq c_{\rm EHZ}(K_2)$. For hyperplane cuts $H$ splitting $K$,

$$c_{\rm EHZ}(K) \leq c_{\rm EHZ}(K_1) + c_{\rm EHZ}(K_2).$$

If $K$ is symplectomorphic to a ball, this inequality is reversed (superadditive), so balls are intrinsically cuts additive. The EHZ capacity is thus central to quantifying action-minimizing properties and understanding dynamical symmetries of non-smooth convex domains (Haim-Kislev, 20 Nov 2025).

3. Topological Zoll Extension: Generalized Zoll Domains

A topological generalization of Zoll domains is formulated using an increasing sequence of Ekeland-Hofer capacities $c_1(K), c_2(K), ...$, satisfying

$0 < c_1(K) \leq c_2(K) \leq \dots.$

On smooth convex domains, these coincide with Gutt–Hutchings capacities. A (possibly non-smooth) convex body $K$ is generalized Zoll if

$c_1(K) = c_n(K).$

Equivalently, the Fadell–Rabinowitz index of the $S^1$-space of action-minimizing closed characteristics is at least $n$.

Two principal theorems clarify the relationship between dynamical and topological characterizations:

• If $K$ is generalized Zoll and the characteristic flow on $\partial K$ is well-defined (each boundary point admits a unique characteristic modulo reparametrization), then $K$ is cuts additive.
• If $K$ is cuts additive and the space of splitting action-minimizers for each cut is contractible, then $K$ is generalized Zoll.

These results establish the equivalence of analytic and topological Zoll extensions in diverse settings (Haim-Kislev, 20 Nov 2025).

4. Definition and Properties of Non-Smooth Action-Minimizing Closed Characteristics

In the non-smooth regime, the notion of closed characteristics relies on convex analysis. For $x \in \partial K$, the normal cone is defined as

$$N_K(x) = \{v : \langle v, x-y \rangle \geq 0\,\,\forall y \in K\}.$$

A generalized characteristic is a loop $\gamma \in W^{1,2}(S^1, \partial K)$ obeying

$-J \dot\gamma(t) \in N_K(\gamma(t)) \text{ a.e.}$

with $J$ the standard complex structure. Loops are normalized via

$$h_K(-J\dot\gamma(t)) = 2, \quad \implies \quad \mathcal{A}(\gamma) = \int_{S^1} \alpha(\dot\gamma) = 1.$$

Variationally, normalized action-minimizing characteristics are minimizers of the functional

$$I_K^p(\gamma) = \left( \frac{1}{2^p} \int_0^1 h_K(\dot\gamma(t))^p\,dt \right)^{2/p}$$

subject to $\mathcal{A}(\gamma) = 1$, all achieving $I_K^p(\gamma) = c_{\rm EHZ}(K)$. The “systole space” is defined as

$\Sys(K) = \{\gamma \in W^{1,2}(S^1, \partial K) : \mathcal{A}(\gamma) = 1,\; I_K^p(\gamma) = c_{\rm EHZ}(K) \}$

and carries a free $S^1$-action via time-shift (Haim-Kislev, 20 Nov 2025).

5. Classification of Local Dynamical Behaviors

For almost every $t \in S^1$ and for every $\gamma \in \Sys(K)$, three types of local dynamics arise:

1. Extreme ray: $-J\dot\gamma(t)$ lies on an extreme ray of $N_K(\gamma(t))$, mirroring the smooth case;
2. Coisotropic face: $\gamma(t)$ traverses a flat face $E \subset \partial K$ whose tangent space is coisotropic (normal cone isotropic), with $-J\dot\gamma(t) \in N_K(E)$ almost everywhere on $E$;
3. Isotropic gliding: $-J\dot\gamma(t)$ remains in the interior of an isotropic subspace of $N_K(\gamma(t))$ of dimension $>1$, but $\gamma$ is not confined to any single coisotropic face.

Isotropic gliding is absent for action-minimizers if $K$ is a convex polytope or a product $K = K_q \times K_p$ of convex Lagrangian bodies. This classification shows that non-smooth dynamics, while potentially more complex, retain significant structure and can avoid certain degenerate behaviors in important cases (Haim-Kislev, 20 Nov 2025).

6. $H^1$-Compactness of the Systole Space modulo Coisotropic Collapse

Loops $\gamma_1, \gamma_2 \in \Sys(K)$ are deemed equivalent, $\gamma_1 \sim \gamma_2$, if they coincide outside the union of all intervals where they respectively slide along some coisotropic face. The quotient space $Q = \Sys(K)/\sim$ is equipped with a pseudo-metric $d_\sim$ induced from the $H^1$-Sobolev norm, defined by infimum over chains of equivalent curves.

If $K$ admits no isotropic gliding for action-minimizers, then the quotient $(Q, d_\sim)$ is compact, and $d_\sim$ agrees with the $C^0$ quotient topology. The compactness is established by showing $C^0$-limits yield $H^1$-convergence off coisotropic intervals, and reparametrizations ensure Cauchy chains descend to the quotient (Haim-Kislev, 20 Nov 2025).

7. Explicit Examples and Boundary Phenomena

Several concrete examples illustrate these constructions:

• The 24-cell in $\mathbb{R}^4$: The regular 24-cell $X$ satisfies $\rho_{\rm sys}(X) = 1$, has no Lagrangian faces, is foliated by minimizers, and is both generalized Zoll and cuts additive.
• Intersection of simplex and cube: For $Y = S \cap Q \subset \mathbb{R}^4$ with $S$ the simplex and $Q = [0, \frac{1}{2}]^4$, $Y$ achieves systolic ratio $1$. Although the characteristic flow is ill-defined at some edges, combinatorial hyperplane cut analysis confirms $Y$ is cuts additive locally, implying it is generalized Zoll.
• Non-additive counterexample: For $P \subset \mathbb{R}^2$ a regular pentagon and $T$ its $90^\circ$-rotation, $P \times T \subset \mathbb{R}^4$ with $\rho_{\rm sys} > 1$ yields a cuts extremizer that is not fully cuts additive, indicating a separation of extremal properties once smoothness is absent (Haim-Kislev, 20 Nov 2025).

These examples delineate the range of possible behaviors in the non-smooth regime, highlighting the necessity and limits of the extended Zoll framework.