Codimension-Zero Tubes: Concepts & Applications

• Codimension-zero tubes are full-dimensional regions in a space, representing either degenerate limits or substantive neighborhoods depending on the theoretical context.
• They span methodologies in areas such as Riemannian geometry, vortex dynamics, complex analysis, and symplectic topology, highlighting their versatile roles in modeling geometric and physical phenomena.
• Applications of codimension-zero tubes include volume invariance in Riemannian settings, regularization of singular supports in general relativity, and the modeling of extension problems in singularity theory.

Codimension-zero tubes are not a single uniform object across the literature. In some theories, the codimension-zero case is a formal limit in which the ordinary tube construction collapses because there are no normal directions; in others, it denotes a genuine full-dimensional neighborhood, collar, domain, or worldtube. The common feature is that the object has the same ambient dimension as the surrounding space, but the mathematical role of that full-dimensionality varies sharply between Riemannian tube formulas, hypoanalytic tube structures, vortex dynamics, Stein and Grauert geometry, singularity theory, laminations, symplectic topology, and general relativity (Cacciatori et al., 2023, Tavares, 2014, Enciso et al., 2012, Forstneric et al., 2018, Savaliya, 28 Jun 2026).

1. Terminological range

Across the cited work, “codimension-zero tube” has several distinct meanings rather than a single standard definition. In Riemannian and hypoanalytic tube theories, codimension zero is effectively a degenerate endpoint of a theory designed for positive codimension. In vortex dynamics, complex analysis, symplectic topology, and gravitation, by contrast, codimension-zero tubes are substantive full-dimensional regions such as solid tori, Runge domains, Grauert tubes, embedded symplectic domains, or smooth matter worldtubes. A further use appears in relative singularity theory, where the relevant objects are codimension-zero submanifolds with boundary, typically collars $V\times[0,\varepsilon]$, from which extension problems are posed (Cacciatori et al., 2023, Enciso et al., 2012, Forstneric et al., 2018, Tanabe, 16 Mar 2026).

Context	Codimension-zero interpretation	Status
Riemannian tube formulas	No normal directions; the tube is just $M$	Degenerate
Tube structures in hypoanalytic geometry	$\mathcal L=\mathbb C\otimes TM$; tube coordinates disappear	Degenerate
Vortex tubes and symplectic domains	Full-dimensional embedded domains	Substantive
Runge and Grauert tubes	Open complex domains biholomorphic to bundle or tangent data	Substantive
Relative Thom theory and laminations	Collars, product neighborhoods, flow boxes	Substantive
Tube percolation	Codimension zero would mean $k=d$, but that case is excluded	Outside setup

2. Degenerate limits in tube theories

In Riemannian tube geometry, the construction is set up for a compact submanifold $M\subset N$ of codimension $q$, with the tube defined by normal geodesics of length $t<\varepsilon$. The Euclidean Weyl formula and the general Jacobi-field volume element are both written in terms of normal data, normal polar coordinates, and the $q$-dimensional normal disc. When $q=0$, all of that structure disappears: the normal fiber is a point, there are no director cosines, no second fundamental form terms, and no nontrivial tube parameter. Interpreted geometrically rather than formally, the tube collapses to $M$ itself, so $M$0, and if $M$1 then $M$2. The same trivialization appears in the concentration-locus formalism: if $M$3, then the tubular neighborhood is already the whole space, the complement has measure $M$4, and the associated Wasserstein, box, and concentration distances collapse immediately (Cacciatori et al., 2023).

The same pattern occurs for Tube structures in the hypoanalytic sense. A Tube structure is a triple $M$5 with $M$6 and $M$7 a commutative Lie algebra satisfying the three displayed conditions

$M$8

If $M$9, then necessarily $\mathcal L=\mathbb C\otimes TM$0. The local defining maps $\mathcal L=\mathbb C\otimes TM$1 are constant, the normal form $\mathcal L=\mathbb C\otimes TM$2 has no $\mathcal L=\mathbb C\otimes TM$3-coordinates, and the local Hartogs criterion becomes automatic because there are no nontrivial fibers $\mathcal L=\mathbb C\otimes TM$4. On the cohomological side, $\mathcal L=\mathbb C\otimes TM$5 is the empty wedge product, hence $\mathcal L=\mathbb C\otimes TM$6, so

$\mathcal L=\mathbb C\otimes TM$7

and the first compactly supported cohomology of the tube complex reduces to ordinary $\mathcal L=\mathbb C\otimes TM$8. The paper treats this as a collapsed, non-genuinely hypoanalytic limit case rather than as substantive Hartogs–Bochner geometry (Tavares, 2014).

3. Full-dimensional geometric realizations

A genuinely codimension-zero usage appears in fluid mechanics. In steady Euler flow, a tube is defined as a bounded domain in $\mathcal L=\mathbb C\otimes TM$9 whose boundary is a smoothly embedded torus; topologically it is a solid torus, diffeomorphic to $k=d$0. The model thin tube around a closed curve $k=d$1 is

$k=d$2

The central result states that for any finite collection of pairwise disjoint, possibly knotted and linked closed curves $k=d$3, and for small enough $k=d$4, there is a diffeomorphism $k=d$5 of $k=d$6, arbitrarily close to the identity in any $k=d$7 norm, such that the images $k=d$8 are vortex tubes of a global Beltrami field $k=d$9 satisfying

$M\subset N$0

Inside each such codimension-zero tube there are uncountably many nested invariant tori, the invariant-torus set has positive Lebesgue measure near the boundary, infinitely many closed vortex lines lie between any two of those tori, and the image $M\subset N$1 is a periodic vortex line (Enciso et al., 2012).

Symplectic topology supplies a different full-dimensional interpretation. There the relevant objects are compact domains $M\subset N$2 and symplectic embeddings

$M\subset N$3

Such an embedding is called unknotted if there exists a symplectomorphism $M\subset N$4 of the target with $M\subset N$5, and knotted otherwise. The paper shows that many toric domains, including $M\subset N$6, admit knotted codimension-zero self-embeddings in this strong sense. Here “tube” does not mean a neighborhood of a lower-dimensional submanifold at all; it means a full-dimensional symplectic region embedded into a larger copy of itself in a manner not straightenable by any ambient symplectomorphism (Gutt et al., 2017).

A boundary-oriented analogue appears in $M\subset N$7-spaces. Horizontal $M\subset N$8-tubes are constant-mean-curvature surfaces invariant under the 1-parameter isometry group generated by a fixed horizontal geodesic and lying at bounded distance from it. They are not defined as equidistant metric tubes, but in the foliation regime they behave as tube boundaries. The family $M\subset N$9 foliates $q$0 when $q$1, and foliates $q$2 when $q$3, under the condition

$q$4

This suggests codimension-zero tube-like regions between successive leaves, even though the primary objects are the CMC boundary surfaces themselves (Manzano, 2023).

4. Complex-analytic tubes and algebraization

In Stein geometry, a Runge tube is a holomorphic embedding of the entire normal bundle $q$5 of a Stein submanifold $q$6 into the ambient Stein manifold $q$7, with image a Runge domain. Since

$q$8

the image is an open domain, hence literally codimension zero in the ambient manifold. The main theorem states that if $q$9 and $t<\varepsilon$0 are Stein, $t<\varepsilon$1, $t<\varepsilon$2 has the density property, and $t<\varepsilon$3 is a holomorphic embedding with $t<\varepsilon$4-convex image, then $t<\varepsilon$5 is approximable uniformly on compacts in $t<\varepsilon$6 by holomorphic embeddings of $t<\varepsilon$7 into $t<\varepsilon$8 whose images are Runge domains. In the algebraic codimension-$t<\varepsilon$9 case, if $q$0 is an algebraic submanifold and $q$1, the normal bundle admits a Runge embedding into $q$2 agreeing with the inclusion on the zero section (Forstneric et al., 2018).

Grauert tubes furnish another codimension-zero complex domain, now attached to a real-analytic Riemannian manifold. For a real-analytic surface $q$3 with Poincaré metric, the Guillemin–Stenzel construction yields a canonical strictly plurisubharmonic function $q$4 on a complexification $q$5, and for small $q$6 the domain

$q$7

is a codimension-zero Grauert tube with strongly pseudoconvex boundary $q$8. In the hyperbolic-surface case, the paper proves that for sufficiently small $q$9 the boundary has nowhere vanishing Cartan CR-curvature, hence no CR-umbilical points. In the upper-half-plane model, the boundary can be written explicitly by a quadratic equation, which is what allows the Cartan-curvature computation (Foo et al., 2019).

Entire Grauert tubes push this viewpoint to the whole tangent bundle $q=0$0. The relevant codimension question there is not the existence of the full tube, but its algebraicity. The paper does not use “codimension-zero tubes” as a technical term; its object is the entire Grauert tube itself, and the main result is a codimension-one approximation to Burns’ conjecture. For an entire Grauert tube $q=0$1, there is a tube singularity $q=0$2, and for any $q=0$3 one finds a holomorphic function $q=0$4 with

$q=0$5

such that

$q=0$6

is biholomorphic to a smooth affine variety. If $q=0$7 and the even cohomology is finite-dimensional, then the whole tube is affine. Thus the codimension-zero object is the ambient entire tube, while the proven algebraization is achieved after removing a hypersurface (Song, 7 May 2026).

5. Collars, prescribed regions, and local product neighborhoods

Relative singularity theory uses codimension-zero submanifolds as prescribed domains for extension problems. The basic datum is a tuple $q=0$8, where $q=0$9 is a compact $M$0-manifold, $M$1 is a union of components of $M$2, $M$3 is an $M$4-manifold, and $M$5 is generic and avoids a specified singularity type $M$6. Extension data $M$7 require that $M$8 and $M$9 are codimension-zero submanifolds, so the singularity locus of an extension lies in the complement $M$00. The standard geometric model is a collar

$M$01

which functions as a tube-like neighborhood of the boundary. The associated relative Thom polynomial

$M$02

is the universal relative cohomological obstruction to extending $M$03 while avoiding $M$04-points (Tanabe, 16 Mar 2026).

Codimension-zero laminations provide a topological analogue. There, codimension zero means that the transversal is a locally compact Polish $M$05-dimensional space, so local neighborhoods are flow boxes of the form

$M$06

or, in compact form, boxes $M$07 with $M$08 clopen and totally disconnected. The paper treats these as the fundamental local product regions, then shows that any codimension-zero lamination is homeomorphic to an inverse limit of branched manifolds and that an inverse limit of branched manifolds is a codimension-zero lamination if and only if the bonding system is flattening. In this setting, codimension-zero tubes are not normal-bundle neighborhoods but foliated product neighborhoods, singular boxes, and inverse-limit local products (Rojo, 2012).

A contrasting exclusion occurs in tube percolation. There the model studies $M$09-dimensional tubes in $M$10 with $M$11, so codimension one corresponds to $M$12. Codimension zero would mean $M$13, i.e. a full-dimensional tube with $M$14-sphere cross-sections, but this case is outside the formal setup. The paper therefore treats codimension-zero tubes not as a degenerate instance, but as an object excluded by definition (Kanazawa et al., 14 May 2026).

6. Gravitating tubes

General relativity supplies perhaps the most explicit program in which codimension-zero tubes are proposed as fundamental objects. The starting point is the Geroch–Traschen obstruction: in Einstein gravity, sources supported on sets of codimension $M$15 are too singular for the nonlinear field equations to admit sufficiently regular metric solutions in the usual distributional framework. A point particle, whose stress-energy is supported on a worldline, is therefore not an acceptable fundamental gravitating source. The proposed replacement is a codimension-zero tube $M$16, a finite spacetime region built inside a tubular neighborhood of an auxiliary timelike curve and foliated by timelike codimension-one hypersurfaces (Savaliya, 28 Jun 2026).

The foliation is generated by a scalar $M$17, with regular leaves $M$18 satisfying $M$19, while the core curve is the Morse–Bott critical set $M$20. The dynamics are governed by the action

$M$21

Variation with respect to $M$22 implies that each leaf has constant mean curvature,

$M$23

while variation with respect to the metric yields a smooth stress-energy tensor supported throughout $M$24, not on a singular lower-dimensional set. For $M$25 and $M$26, the null, weak, and dominant energy conditions remain satisfied while the strong energy condition is violated. In the ultraviolet limit, where the tube radius tends to zero and the Lagrangian density is rescaled appropriately, the action reduces to the point-particle action together with a canonical self-force-like term, and the particle rest mass emerges as an effective quantity rather than a fundamental localized parameter. The perturbation analysis further shows infinite squared sound speed, so the foliation-generating scalar is cuscuton-like, and normal deformations of the leaves satisfy the Jacobi equation for timelike hypersurface congruences (Savaliya, 28 Jun 2026).

Taken together, these examples indicate two persistent patterns. In some theories, codimension-zero tubes are merely the vacuous endpoint of a positive-codimension formalism. In others, they are the natural full-dimensional replacement for singular supports, lower-dimensional cores, or purely local neighborhoods. This suggests that the mathematical significance of codimension-zero tubes is not their topology alone, but the shift from transverse thickening to genuinely volumetric geometry.