Minimal Horizontal Triods in the Heisenberg Group

• Minimal horizontal triods are Steiner-type minimizing networks in the first Heisenberg group, defined by three horizontal curves meeting at a common junction with area constraints.
• Their formulation reduces the spatial problem to a planar network where horizontal length equals the Euclidean length of the projection and vertical displacement is governed by a signed-area functional.
• The variational framework employs constant-curvature conditions and a projected 120° balance at the junction, yet exhibits nonuniqueness and potential degenerate configurations.

Minimal horizontal triods are Steiner-type minimizing networks in the first Heisenberg group $\mathcal H$: they consist of three horizontal curves with prescribed endpoints, meeting at a common junction and minimizing the total horizontal length. Their geometry is governed by a characteristic sub-Riemannian reduction: horizontal length is exactly the Euclidean length of the planar projection, while the vertical coordinate is constrained by a signed-area functional of that projection. As a consequence, the relevant minimization problem is neither the classical Euclidean Steiner tree problem nor an unconstrained geodesic-network problem, but a planar network problem with nonholonomic area constraints. The current analytic formulation includes existence of possibly degenerate minimizers, a variational characterization of critical networks, a horizontal curve-shortening flow, and a stable fully discrete finite element scheme used to explore the solution landscape (Nürnberg et al., 21 Jul 2025).

1. Heisenberg geometry and horizontality

The ambient space is the first Heisenberg group

$\mathcal H=(\mathbb R^3,\circ),$

with group law

$$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$

Its left-invariant vector fields are

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$

with $[X_1,X_2]=X_3$. The horizontal distribution at $p\in\mathcal H$ is the $2$-plane $\mathrm{span}\{X_1(p),X_2(p)\}$. The metric used in the variational problem is obtained by declaring $\{X_1(p),X_2(p),X_3(p)\}$ orthonormal and then restricting to the horizontal distribution (Nürnberg et al., 21 Jul 2025).

A sufficiently smooth curve $\gamma:[0,1]\to\mathcal H$ is horizontal if $\mathcal H=(\mathbb R^3,\circ),$0 for all $\mathcal H=(\mathbb R^3,\circ),$1. Writing $\mathcal H=(\mathbb R^3,\circ),$2, the horizontality constraint is

$\mathcal H=(\mathbb R^3,\circ),$3

Equivalently,

$\mathcal H=(\mathbb R^3,\circ),$4

Thus the $\mathcal H=(\mathbb R^3,\circ),$5-coordinate is not free: it is determined by the planar projection $\mathcal H=(\mathbb R^3,\circ),$6.

For a horizontal curve,

$\mathcal H=(\mathbb R^3,\circ),$7

so the horizontal length is

$\mathcal H=(\mathbb R^3,\circ),$8

Hence the sub-Riemannian length is exactly the Euclidean length of the planar projection. The associated signed-area functional is

$\mathcal H=(\mathbb R^3,\circ),$9

and for a horizontal curve joining $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$0 to $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$1,

$$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$2

This identity is the basic structural mechanism behind minimal horizontal triods: vertical displacement is encoded by planar oriented area rather than by ordinary Euclidean motion in the $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$3-direction.

2. Definition of a minimal horizontal triod

Given three distinct points $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$4, $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$5, a horizontal triod is a triple

$$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$6

of regular horizontal curves $$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$7 such that

$$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$8

and

$$\begin{pmatrix} x\ y\ z \end{pmatrix}\circ \begin{pmatrix} \hat x\ \hat y\ \hat z \end{pmatrix} = \begin{pmatrix} x+\hat x\ y+\hat y\ z+\hat z+\frac12(x\hat y-y\hat x) \end{pmatrix}.$$9

The common point $$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$0 is the junction. The total length is

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$1

A minimal horizontal triod is a horizontal triod minimizing this total horizontal length among all admissible triods connecting the same three points (Nürnberg et al., 21 Jul 2025).

The theory allows degeneracy. A triod is degenerate if the junction coincides with one endpoint, say $$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$2, and the corresponding curve collapses to a point. Equivalently, in projection one component is constant. Because the three spatial points are distinct, at most one branch can collapse.

The minimization problem is reduced to the plane. Let

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$3

These are the admissible planar networks, with common projected junction

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$4

The planar length is

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$5

The horizontality constraints are encoded by the equalities

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$6

where

$$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$7

Thus the spatial problem is equivalent to minimizing $$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$8 over $$X_1=\frac{\partial}{\partial x}-\frac y2\frac{\partial}{\partial z},\qquad X_2=\frac{\partial}{\partial y}+\frac x2\frac{\partial}{\partial z},\qquad X_3=\frac{\partial}{\partial z},$$9 under these area constraints.

A useful parametrization concept is quasinormality. A curve $[X_1,X_2]=X_3$0 is quasinormal if

$[X_1,X_2]=X_3$1

A triod is quasinormal if each branch is quasinormal; in the degenerate case, the two nonconstant branches are required to be quasinormal. This constant-speed structure arises by reparametrization and is used in the compactness and Euler–Lagrange analysis.

3. Existence and variational characterization

Existence is proved by the direct method after adding a Dirichlet regularization. One introduces

$[X_1,X_2]=X_3$2

and

$[X_1,X_2]=X_3$3

For $[X_1,X_2]=X_3$4, the regularized problem is

$[X_1,X_2]=X_3$5

The minimizers of this problem are shown to be constant-speed on each branch, and uniform estimates independent of $[X_1,X_2]=X_3$6 allow passage to the limit $[X_1,X_2]=X_3$7. The resulting theorem states that for any three distinct points $[X_1,X_2]=X_3$8, there exists a minimal horizontal triod, possibly degenerate (Nürnberg et al., 21 Jul 2025).

For a single horizontal curve connecting $[X_1,X_2]=X_3$9 to $p\in\mathcal H$0, the projected minimization problem is

$p\in\mathcal H$1

The first variation with a Lagrange multiplier $p\in\mathcal H$2 yields

$p\in\mathcal H$3

and hence constant planar curvature,

$p\in\mathcal H$4

Therefore a minimizing horizontal curve projects either to a straight line ($p\in\mathcal H$5) or to a circular arc ($p\in\mathcal H$6).

The same mechanism governs triods. For a nondegenerate minimizer $p\in\mathcal H$7, there are multipliers $p\in\mathcal H$8 such that branchwise

$p\in\mathcal H$9

Consequently,

$2$0

Each projected branch is therefore a smooth curve of constant curvature, hence a line segment or circular arc. At the junction, the natural boundary terms give

$2$1

which is the classical vector balance relation for equal surface tensions. In the plane this means that the three projected tangents meet at $2$2.

The degenerate case is different. If one branch collapses, then the two remaining projected branches still satisfy constant-curvature equations, but there is no genuine triple-junction condition and hence no $2$3 law. This sharply distinguishes the Heisenberg problem from the classical Euclidean Steiner tree problem: the familiar Euclidean dichotomy between a $2$4 Steiner point and a vertex-degenerate tree does not transfer unchanged, because the area constraints can force circular-arc geometry and can produce nontrivial degenerate minimizers (Nürnberg et al., 21 Jul 2025).

4. Nonuniqueness and sub-Riemannian geometry

Uniqueness generally fails. The analysis explicitly notes that uniqueness cannot be expected, since even horizontal geodesics between two points in the Heisenberg group need not be unique, especially when the points have the same planar projection. Minimal triods inherit this nonuniqueness. A particularly instructive family occurs when the endpoints lie on the $2$5-axis: in that case the rotational symmetry around the $2$6-axis can generate infinitely many equal-length minimizing or critical configurations (Nürnberg et al., 21 Jul 2025).

The geometric source of this behavior is the area constraint

$2$7

which converts a spatial endpoint condition into a planar integral constraint. In Euclidean Steiner theory, a minimizing branch is a geodesic segment and the network is determined largely by angle balance. In the Heisenberg setting, by contrast, minimizing branches are determined by a curvature-plus-area relation. The projected network behaves like a constrained curvature network: straight segments appear only when the relevant Lagrange multiplier vanishes, whereas generic nonzero multipliers produce circular arcs.

This also explains why the classical Euclidean threshold criterion based on an angle $2$8 does not persist in any simple form. The numerical experiments show that in obtuse projected triangles the horizontal flow often does not produce degeneration, while in more extreme cases the shortest branch can shrink to zero and the computation encounters a singularity. The data therefore support the conclusion that there is no clear simple threshold criterion based only on the projected triangle geometry (Nürnberg et al., 21 Jul 2025).

For nondegenerate minimizers, the projected $2$9 rule survives exactly, but only in projection. In space, there is no simple full $\mathrm{span}\{X_1(p),X_2(p)\}$0-dimensional angle law analogous to Euclidean Steiner trees in $\mathrm{span}\{X_1(p),X_2(p)\}$1, because the admissible tangent directions are constrained by horizontality and the vertical coordinate is slaved to the planar area. A plausible implication is that the Heisenberg Steiner problem is best viewed not as a perturbation of Euclidean network geometry in $\mathrm{span}\{X_1(p),X_2(p)\}$2, but as a planar network problem with hidden nonholonomic constraints.

5. Horizontal curve-shortening flow and computation

To explore the minimizing landscape, the theory introduces a horizontal curve-shortening flow. For a single curve, the flow is

$\mathrm{span}\{X_1(p),X_2(p)\}$3

with fixed endpoints and

$\mathrm{span}\{X_1(p),X_2(p)\}$4

The $\mathrm{span}\{X_1(p),X_2(p)\}$5-term is essential: it preserves horizontality during the evolution (Nürnberg et al., 21 Jul 2025).

For triods, the projected formulation is more transparent. If $\mathrm{span}\{X_1(p),X_2(p)\}$6 and

$\mathrm{span}\{X_1(p),X_2(p)\}$7

then the projected strong flow is

$\mathrm{span}\{X_1(p),X_2(p)\}$8

$\mathrm{span}\{X_1(p),X_2(p)\}$9

$\{X_1(p),X_2(p),X_3(p)\}$0

together with fixed projected endpoints, a common projected junction, and the balance law

$\{X_1(p),X_2(p),X_3(p)\}$1

The multipliers $\{X_1(p),X_2(p),X_3(p)\}$2 are chosen so that the area quantities $\{X_1(p),X_2(p),X_3(p)\}$3 remain compatible with horizontal lifting.

The total length decreases along smooth solutions: $\{X_1(p),X_2(p),X_3(p)\}$4 This gives a genuine energy-dissipation identity, but the available results do not include long-time existence or convergence of the continuous flow to a minimizer.

The computational treatment uses a weak formulation and a fully discrete finite element scheme on piecewise affine spaces. Under a mild nondegeneracy assumption, the scheme has a unique solution at each time step and is unconditionally stable. Its discrete stability estimate is

$\{X_1(p),X_2(p),X_3(p)\}$5

This mirrors the continuous dissipation law and makes the scheme suitable for numerical exploration of a landscape that is visibly multi-well and nonunique (Nürnberg et al., 21 Jul 2025).

The experiments isolate several recurrent phenomena. When all $\{X_1(p),X_2(p),X_3(p)\}$6-coordinates vanish, the classical planar Steiner configuration of three straight segments meeting at $\{X_1(p),X_2(p),X_3(p)\}$7 is stationary. In obtuse-triangle regimes, degeneration is not governed by a simple Euclidean angle threshold. When all three endpoints have the same planar projection, the limiting projected networks can resemble standard double-bubble-type or unequal-area double-bubble configurations; branch disappearance, metastable rearrangements, and lens-type steady states also occur. In addition, computations exhibit singularity formation by branch collapse and families of degenerate critical configurations with continuous rotational nonuniqueness.

6. Related notions and terminological disambiguation

The phrase “minimal horizontal triods” names a specific sub-Riemannian problem in the Heisenberg group. Several nearby literatures use some of the same words but refer to different objects.

Paper	Main object	Relation to minimal horizontal triods
"Minimal horizontal triods: Analysis and computation" (Nürnberg et al., 21 Jul 2025)	Horizontal Steiner-type networks in $\{X_1(p),X_2(p),X_3(p)\}$8	Direct treatment
"Forcing minimal patterns of triods" (Bhattacharya, 2021)	Periodic patterns on the $\{X_1(p),X_2(p),X_3(p)\}$9-od and forcing order	Not a junction-minimization problem
"Uniform A Priori Estimates For A Class Of Horizontal Minimal Equations" (Earp, 2012)	Horizontal minimal graph equations in $\gamma:[0,1]\to\mathcal H$0	Analytic control for graph pieces, not triods
"Free boundary minimal surfaces in the unit 3-ball" (Folha et al., 2015)	Doubled horizontal disks joined by catenoidal bridges	Nearby free-boundary gluing picture, not triods
"Minimal networks on balls and spheres for almost standard metrics" (Sciaraffia, 2024)	Minimal geodesic triods in $\gamma:[0,1]\to\mathcal H$1 for almost standard metrics	Riemannian analogue without Heisenberg horizontality

The triod theory in the $\gamma:[0,1]\to\mathcal H$2-od literature concerns a branched one-dimensional topological space

$\gamma:[0,1]\to\mathcal H$3

and studies periodic patterns, forcing, rotation numbers, and the class of minimal same-rotation patterns called triod twists. This is a combinatorial-dynamical notion of “triod,” not a minimizing network of horizontal curves (Bhattacharya, 2021).

The PDE literature on horizontal minimal equations addresses smooth positive horizontal graphs in $\gamma:[0,1]\to\mathcal H$4, proving $\gamma:[0,1]\to\mathcal H$5, boundary $\gamma:[0,1]\to\mathcal H$6, and modulus-of-continuity estimates, and in dimension $\gamma:[0,1]\to\mathcal H$7 an existence theorem for the Dirichlet problem. Those results concern single-valued graph pieces rather than a singular Y-junction, although they may be relevant for branchwise analysis away from a junction (Earp, 2012).

The free-boundary surface construction in the unit ball is likewise distinct. It produces, for sufficiently large $\gamma:[0,1]\to\mathcal H$8, embedded free boundary minimal surfaces obtained by doubling the horizontal disk and inserting small half-catenoidal bridges near the boundary and, in the genus-one family, a central catenoidal neck. The limiting geometry is a double copy of the unit horizontal disk or punctured disk, not a triod. The paper is positioned relative to the earlier existence result of A. Fraser and R. Schoen for genus-zero free boundary minimal surfaces with $\gamma:[0,1]\to\mathcal H$9 boundary components (Folha et al., 2015).

A closer classical analogue is the Riemannian theory of minimal triods in the unit ball with a metric close to Euclidean. There the standard model consists of three geodesic edges meeting at the origin with equal $\mathcal H=(\mathbb R^3,\circ),$00 angles and orthogonal contact at the boundary; under small $\mathcal H=(\mathbb R^3,\circ),$01-perturbations of the metric, minimal triods persist, and multiplicity is obtained via a finite-dimensional reduction on the Stiefel manifold $\mathcal H=(\mathbb R^3,\circ),$02 and Lusternik–Schnirelmann category. This provides a geodesic-network counterpart to the Heisenberg problem, but without the area-constrained horizontal geometry that forces circular-arc branches and nonholonomic compatibility conditions (Sciaraffia, 2024).

In this broader landscape, minimal horizontal triods occupy a specific position: they are minimal networks, but their variational structure is controlled by sub-Riemannian horizontality. The decisive feature is the identity between vertical displacement and projected signed area, which transforms the classical Steiner problem into a constrained planar curvature-network problem and produces the characteristic combination of projected $\mathcal H=(\mathbb R^3,\circ),$03 balance, constant-curvature branches, degeneracy, and pronounced nonuniqueness (Nürnberg et al., 21 Jul 2025).