Maxfaces with infinitely many Swallowtails

Abstract: In this article, we discuss the existence of a 1-parameter infinite genus family of maxfaces having infinitely many planar (spacelike) ends and infinitely many swallowtails. In particular, we show the existence of the following: (1) a period-2 family of maxfaces with infinitely many planar ends and alternating singularity types, where every odd-layer neck has exactly four swallowtails, while each even-layer neck is almost conical; for $n=2$, the fundamental piece is a genus-1 Wei-type maxface; (2) a period-3 family where every neck carries four swallowtails (24 per period); and (3) a period-2 family of maxfaces with an almost conical singularity on every neck. All maxfaces are embedded in a wider sense.

• The paper constructs explicit infinite-genus maxfaces using a systematic node-opening technique to manage complex singular arrangements.
• It introduces two singular types, standard swallowtails with computed residue invariants and almost-conical points identified via vanishing Taylor invariants.
• The study extends finite-genus methods to infinite concatenations, offering new insights into singular maximal surfaces in Lorentzian geometry.

Infinite-Genus Maxfaces with Infinitely Many Swallowtails and Almost-Conical Singularities

Introduction and Background

This work establishes the existence and explicit construction of 1-parameter families of complete, embedded (in the sense of being embedded outside a compact set) maximal surfaces (maxfaces) in the Lorentz-Minkowski 3-space $\mathbb{E}^{3}_{1}$, distinguished by their infinite genus, infinitely many planar ends, and a singular set comprising both infinitely many swallowtails and almost-conical singularities. The problem is motivated by the rigidity of complete maximal surfaces in $\mathbb{E}^{3}_{1}$, where the only smooth, complete example is the spacelike plane. By allowing controlled singularities—following the framework of Umehara and Yamada—the class of admissible surfaces is enriched, notably permitting complete non-planar examples.

Recent progress in the finite-genus regime, notably the node-opening and gluing techniques adapted by Chen, Dhochak, Kumar, and Mohanty to the Lorentzian context, yields wide families of maxfaces with arbitrary genus and many ends, provided the combinatorics satisfy suitable balance and non-degeneracy conditions. The present work extends these methodologies, constructing infinite-genus examples by a systematic concatenation of finite configurations, subject to careful modifications ensuring analyticity, proper handling of singular sets, and resolution of the period problem.

Construction Methodology

The constructions rely on an infinite chain of Riemann spheres, equipped with precise Weierstrass data $(g, dh)$ tailored to the maxface context. The surfaces are parameterized by a configuration $(p_{k,i})_{k\in\mathbb{Z},\,1\leq i\leq n_k}$ dictating the location of necks, with types encoded by the sequence $(n_k)_{k\in \mathbb{Z}}$. The gluing is effected via the generalized node-opening technique, ensuring the compatibility of the period problems across infinitely many levels.

Configurations are balanced (the sum of forces vanishes) and non-degenerate (the relevant Jacobian is invertible on $l^\infty(\mathbb{Z})$). Special attention is given to periodic (or quasi-periodic) configurations, enabling explicit computation and direct control of singular behavior. The surfaces thus obtained are periodic (of small period, typically 2 or 3), with infinite genus, infinitely many planar (spacelike) ends, and a highly structured singular set.

The construction of the Weierstrass data ensures that the immersion is spacelike away from the singular set (where $|g|\neq 1$), and the geometry of the singular set can be directly traced to the behavior of the Gauss map and residues corresponding to the configuration of necks.

Analysis of Singularities

The singularities of the constructed surfaces are located on the "waists" of the necks, each determined by a function $R^{(r)}_{k,i}(\theta)$, a variant of the residue-based invariants from the finite-genus setting, now adapted for the infinite concatenation. A neck is generically adorned with four swallowtail singularities (detected by the nonvanishing of $R^{(1)}_{k,i}$) per layer, as in the case of minimal and maximal surfaces with finite genus.

A major innovation is the identification and rigorous characterization of "almost-conical" singularities. These appear when the entirety of the Taylor expansion invariants $R^{(r)}_{k,i}(\theta)$ vanish for all $\mathbb{E}^{3}_{1}$0 and $\mathbb{E}^{3}_{1}$1—the singular locus persists for $\mathbb{E}^{3}_{1}$2, but the profile of the singularity lacks any angular dependence, behaving like a degenerate, highly isotropic conical point. This phenomenon is not observable in the finite-genus case and is intrinsic to the infinite, periodic stacking of configurations. The work provides an explicit criterion for such singularities and proves their existence in specific layers within the constructed families.

Explicit Families and Strong Results

Several family types are exhibited:

1. Period-2 Families (Alternating Singularities): These have alternating layers where necks produce either standard swallowtails (exactly four per neck) or are almost-conical. For $\mathbb{E}^{3}_{1}$3, the fundamental domain is a genus-1 Wei-type maxface; for general $\mathbb{E}^{3}_{1}$4, the genus rises accordingly. The alternating pattern is controlled by explicit computation of the residue invariants.
2. Period-3 Families (Uniform Swallowtail Proliferation): Every neck in all layers carries four swallowtail singularities, yielding 24 swallowtails per periodic block and thus infinitely many in total. The explicit configuration ensures balanced non-degenerate gluing and symmetry.
3. Period-2 Families with Only Almost-Conical Singularities: All necks in these examples have almost-conical singularities exclusively (i.e., all invariants vanish), providing a sharp contrast with the alternating or fully swallowtail models. These geometrically mimic the classical, periodic Riemann-type maximal surfaces in their Lorentzian form.

In each case, the construction is explicit, and the singularity structure is computed in detail, with all assertions backed by verifiable residue calculations and analytic continuation.

Implications and Future Directions

This research advances both the theoretical landscape and the constructive methodology in the study of maximal surfaces in Lorentz-Minkowski space. By pushing node-opening techniques to the infinite-genus regime under strict analytical control (finiteness hypotheses, balance/non-degeneracy), the paper clarifies the structure of admissible singular sets for maxfaces, revealing novel behaviors (almost-conical points) absent in prior work.

Practically, these families serve as a testbed for further exploration of geometric, analytic, and topological properties of singular maximal surfaces. The approach interfaces smoothly with ongoing work on periodic and quasi-periodic minimal surfaces and could inspire analogues in other curved or higher-dimensional Lorentzian settings.

Open directions include a full classification of singularities arising in infinite concatenations, exploration of moduli spaces of such surfaces, geometric flow problems (mean curvature flow with singularities), and further connections with integrable systems, holomorphic curve theory, and relativistic field equations.

Conclusion

The article presents rigorous construction and computation for infinite-genus maxfaces with infinitely many planar ends and infinitely many singularities of two distinguished types: swallowtails and almost-conical points. The main results offer a comprehensive framework for maximizing families with rich, explicit singular sets, highlight the interaction between combinatorics of configuration, periodicity, and singular geometry, and set the stage for further analytic and geometric investigations in the theory of maximal surfaces in Lorentzian geometry.