Free Boundary Flat Y-Cone Model

• The free boundary flat Y-cone is a singular Plateau model consisting of three planar half-unit disks that meet along a common diameter at 120° angles with free-boundary conditions on the unit sphere.
• Its configuration is rigidly maintained through conformal minimal immersion, balanced junction conditions, and orthogonal free-boundary constraints that enforce planarity on each face.
• The model is characterized by a Morse index of two and a five-dimensional nullity, highlighting its stability and serving as a key reference in free-boundary minimal surface theory.

The free boundary flat $Y$-cone is the standard singular Plateau model in which three planar half-unit disks meet along a common diameter at equal angles of $120^\circ$, while the outer boundary lies on the unit sphere and is orthogonal to it. In current arXiv usage, this object is most precisely the compact model denoted $YC$ in the study of free-boundary minimal $Y$-surfaces in $B^3$, where rigidity, index, and classification results are now available (Matinpour, 29 Sep 2025). It must be distinguished from several different “free boundary cone” problems on arXiv—especially one-phase Bernoulli problems on singular ambient cones and capillary hypersurface cones—which concern cone-tip interaction or codimension-one capillary geometry rather than triple-junction $Y$-surfaces (Allen, 2017).

1. Geometric model and terminology

The ambient space for the classical flat $Y$-cone problem is the unit ball

$$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$

A free boundary minimal surface in $B^3$ is a minimal surface $\Sigma\subset B^3$ whose boundary lies on $120^\circ$0 and which meets $120^\circ$1 orthogonally along $120^\circ$2 (Matinpour, 29 Sep 2025).

In the singular setting, the surface is a triple junction surface

$120^\circ$3

where each $120^\circ$4 is a smooth two-sided surface, the identified boundary components $120^\circ$5 are glued together along a common smooth curve $120^\circ$6, and the remaining components $120^\circ$7 form the outer boundary

$120^\circ$8

A minimal $120^\circ$9-surface in $YC$0 means that each face $YC$1 is minimally immersed, the singular set is the common junction curve $YC$2, and at each $YC$3 the three outward conormals $YC$4 satisfy

$YC$5

This is the standard balancing law for a $YC$6-junction (Matinpour, 29 Sep 2025).

The standard compact flat $YC$7-cone is built from the half-disk

$YC$8

with boundary decomposition

$YC$9

where

$Y$0

The model domain is

$Y$1

where $Y$2 denotes rotation of $Y$3 by angle $Y$4 about the $Y$5-axis. Geometrically, $Y$6 is the union of three planar half-unit disks meeting along their diameters at equal $Y$7 angles. It is “flat” because each face is planar, so

$Y$8

The singular set is the common diameter, hence in the compact model a line segment with endpoints on $Y$9 (Matinpour, 29 Sep 2025).

Along the junction, the paper also imposes the metric compatibility condition

$B^3$0

where

$B^3$1

This is the intrinsic counterpart of the $B^3$2 balancing relation (Matinpour, 29 Sep 2025).

2. Conformal minimal immersion and free-boundary structure

A map

$B^3$3

is conformal when each restriction $B^3$4 is conformal on its face and the three maps agree along the common edge: $B^3$5 On each face, conformality and minimality are expressed by

$B^3$6

In polar coordinates $B^3$7 on $B^3$8, these become

$B^3$9

(Matinpour, 29 Sep 2025).

The free-boundary condition along the outer semicircle $Y$0 is that $Y$1 and the surface meets the sphere orthogonally. Analytically,

$Y$2

for some scalar function $Y$3, and differentiation yields

$Y$4

This identity is one of the main inputs in the rigidity argument (Matinpour, 29 Sep 2025).

Along the singular diameter $Y$5, the triple-junction conditions are encoded by

$Y$6

$Y$7

and

$Y$8

Using harmonicity, the paper further derives

$Y$9

(Matinpour, 29 Sep 2025).

These formulas place the flat $Y$0-cone in a highly rigid class: the conformal structure is sectorwise classical, but the singular geometry is carried by the matching, balancing, and free-boundary orthogonality constraints. A plausible implication is that the object is best understood as a Plateau singularity model with analytic control on each face rather than as a weak multi-phase interface.

3. Rigidity theorem and planarity of the faces

The core rigidity statement is that any conformal and minimal immersion of the standard compact flat $Y$1-cone into $Y$2 meeting $Y$3 orthogonally is again a flat $Y$4-cone (Matinpour, 29 Sep 2025). A later paper restates the earlier theorem in the broader form that any conformal minimal immersion of the flat $Y$5-cone into $Y$6 that meets the boundary sphere orthogonally must coincide with the flat $Y$7-cone itself, up to orthogonal transformation (Matinpour, 26 May 2026).

The proof adapts Nitsche’s complex-analytic method from the disk case. For each face one defines

$Y$8

and then

$Y$9

Because each $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$0 is harmonic and conformal,

$$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$1

is holomorphic. The free-boundary identity $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$2 shows that $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$3 is real on $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$4, while the matching and balancing conditions imply that $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$5 on $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$6. Hence $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$7 is holomorphic and real on the whole boundary, so it is constant; since it vanishes at the origin,

$$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$8

From this one obtains along $$B^3=\mathbb B^3\subset \mathbb R^3,\qquad \partial B^3=\mathbb S^2.$$9

$B^3$0

(Matinpour, 29 Sep 2025).

The vanishing of the quartic differential forces the boundary curves $B^3$1 to be arcs of great circles on $B^3$2. If $B^3$3 is the plane containing $B^3$4, with unit normal $B^3$5, then $B^3$6 is harmonic and vanishes on $B^3$7. By the free-boundary condition,

$B^3$8

and Calderón unique continuation yields

$B^3$9

Therefore each face lies in a plane, and the image is the planar $\Sigma\subset B^3$0-configuration (Matinpour, 29 Sep 2025).

The later $\Sigma\subset B^3$1-cone rigidity paper describes this as the prototype singular rigidity mechanism: conformal parametrization on each sector, harmonicity and conformality, free-boundary orthogonality, junction balance, a holomorphic quadratic differential with real boundary values, vanishing of that differential, and planarity of the faces (Matinpour, 26 May 2026).

4. Morse index, nullity, and the index-two characterization

The flat $\Sigma\subset B^3$2-cone is not only rigid under conformal free-boundary minimal immersion; it also occupies a distinguished place in the Morse-index landscape. The paper proves:

The Morse index of the free boundary flat $\Sigma\subset B^3$3-cone in the unit ball is two, and its nullity is five. Moreover, if $\Sigma\subset B^3$4 is a free boundary minimal $\Sigma\subset B^3$5-surface in the unit ball with Morse index two, then $\Sigma\subset B^3$6 is a $\Sigma\subset B^3$7-cone (Matinpour, 29 Sep 2025).

Normal variations are written as

$\Sigma\subset B^3$8

with admissible space

$\Sigma\subset B^3$9

The compatibility condition

$120^\circ$00

preserves the $120^\circ$01-junction under variation (Matinpour, 29 Sep 2025).

The second variation quadratic form is

$120^\circ$02

Since each face is flat,

$120^\circ$03

and because the junction is straight in the flat model,

$120^\circ$04

Along the outer free boundary on $120^\circ$05,

$120^\circ$06

Thus the spectral problem reduces to harmonic functions on the half-disk with Steklov boundary conditions (Matinpour, 29 Sep 2025).

The relevant Steklov spectrum on each face is

$120^\circ$07

and for $120^\circ$08,

$120^\circ$09

For compatible triples with coefficients $120^\circ$10 satisfying

$120^\circ$11

the $120^\circ$12 eigenspace contributes a $120^\circ$13-dimensional negative space, giving index $120^\circ$14, while the $120^\circ$15 eigenspace yields a $120^\circ$16-dimensional kernel, giving nullity $120^\circ$17 (Matinpour, 29 Sep 2025).

The index-two classification uses ambient translations. If $120^\circ$18 are parallel vector fields in $120^\circ$19, their normal components are Jacobi fields on each face. The paper shows that whenever such a normal component is nontrivial,

$120^\circ$20

Since there are three independent ambient translation directions but the index is only $120^\circ$21, one direction must have vanishing normal component everywhere. That forces an ambient direction tangent to all three faces, hence tangent to the junction curve $120^\circ$22, which must therefore be a straight line segment. From this the paper concludes that the surface is a flat $120^\circ$23-cone (Matinpour, 29 Sep 2025).

5. Position within the Plateau-model rigidity theory

The flat $120^\circ$24-cone now appears as one of the three canonical free-boundary Plateau model surfaces in the unit ball: the planar disk, the flat $120^\circ$25-cone, and the flat $120^\circ$26-cone (Matinpour, 26 May 2026). In the language of Plateau singularities, a surface is locally modeled on one of

$120^\circ$27

where

$120^\circ$28

This is the local model for $120^\circ$29-junctions, and tangent cones at $120^\circ$30-points are required to be $120^\circ$31 up to orthogonal transformation (Matinpour, 26 May 2026).

The unifying rigidity statement is:

Let $120^\circ$32 be a free-boundary minimal Plateau surface in $120^\circ$33 arising as the image of a conformal minimal immersion of one of the classical model domains: a planar disk, the flat $120^\circ$34-cone, or the flat $120^\circ$35-cone. Then $120^\circ$36 is congruent, via an orthogonal transformation of $120^\circ$37, to the corresponding flat model (Matinpour, 26 May 2026).

Within this trichotomy, the flat $120^\circ$38-cone is the intermediate singular model between the smooth disk and the tetrahedral $120^\circ$39-cone. The later $120^\circ$40-cone paper explicitly states that its proof “follows the same strategy as the uniqueness proof for the flat $120^\circ$41-cone,” adapted to tetrahedral combinatorics. This places the $120^\circ$42-cone theorem in a structural role: it is both a classification result and a methodological prototype for singular free-boundary Plateau rigidity (Matinpour, 26 May 2026).

A plausible implication is that the flat $120^\circ$43-cone should be regarded as the first singular analogue of Nitsche’s equatorial disk: the singularity is permitted, but conformality, minimality, and the free-boundary condition still force the model to be planar.

6. Distinct cone problems and common misidentifications

The phrase “free boundary cone” has several incompatible meanings in current literature. The flat $120^\circ$44-cone belongs to the Plateau/minimal-surface setting, not to the one-phase Bernoulli or capillarity settings. The distinction is substantive rather than terminological.

Setting	Model object	Relation to the flat $120^\circ$45-cone
Free-boundary minimal $120^\circ$46-surface	Three planar half-disks meeting at $120^\circ$47 in $120^\circ$48	Genuine flat $120^\circ$49-cone (Matinpour, 29 Sep 2025)
One-phase Bernoulli on a right circular cone	Rotationally symmetric $120^\circ$50-homogeneous free boundary on a singular ambient cone	Not a $120^\circ$51-junction (Allen, 2017)
Capillary or wedge free-boundary hypersurfaces	Codimension-one hypersurface with contact-angle condition	Generally excludes triple-junction $120^\circ$52-cones (Pacati et al., 11 Feb 2025)

In the one-phase Bernoulli problem on the three-dimensional right circular cone

$120^\circ$53

the governing system is

$120^\circ$54

and the relevant homogeneous solution is a rotationally symmetric cone

$120^\circ$55

The authors state explicitly that this problem is “not about classical triple-junctions or soap-film $120^\circ$56-cones,” and that the resulting cone-tip blow-up is a single conical sheet rather than a $120^\circ$57-shape. In dimension three, stable homogeneous solutions are, up to rotation, exactly the symmetric solution $120^\circ$58, and the structural lemmas force the zero set to be a single connected convex cone in a half-space, which is incompatible with a genuine $120^\circ$59-junction (Allen, 2017). The later computational refinement proves that for

$120^\circ$60

the distinguished homogeneous cone solution is the unique minimizer for its boundary data, while the stability threshold is numerically

$120^\circ$61

again in a setting unrelated to flat $120^\circ$62-cones (Allen et al., 2021).

A different but related dictionary appears in the correspondence between free boundary minimal surfaces in $120^\circ$63 and homogeneous one-phase free boundary cones in $120^\circ$64. There the cone problem is

$120^\circ$65

with $120^\circ$66 homogeneous of degree $120^\circ$67. The paper proves that if $120^\circ$68 is diffeomorphic to a disk, then $120^\circ$69 is a half-space; if $120^\circ$70 is diffeomorphic to an annulus, then $120^\circ$71 is a circular cone formed by lines with aperture

$120^\circ$72

This again classifies smooth one-phase cone geometries but does not address triple-junction $120^\circ$73-cones (Nadirashvili et al., 2018).

Capillarity and wedge problems impose yet another framework. For minimizing capillary cones in the half-space, one studies

$120^\circ$74

with stability inequality

$120^\circ$75

In this class, the paper proves that in dimension $120^\circ$76 any minimizing capillary cone with non-sign-changing $120^\circ$77 is flat, and that axially symmetric minimizing capillary cones are flat for $120^\circ$78. But it also emphasizes that a classical $120^\circ$79-cone is generally outside the framework because the objects are boundaries of a single set of finite perimeter rather than multi-sheet triple-junction surfaces (Pacati et al., 11 Feb 2025).

The same exclusionary pattern appears in wedge free-boundary minimal hypersurfaces. In a $120^\circ$80-dimensional wedge $120^\circ$81, $120^\circ$82, any stable $120^\circ$83-to-edge properly embedded free boundary minimal hypersurface $120^\circ$84 is flat: $120^\circ$85 for some hyperplane $120^\circ$86. This is a strong flatness theorem, but its admissible class is properly embedded hypersurfaces, not $120^\circ$87-junction cones (Yan, 2024).

The most precise conclusion is therefore terminological and geometric at once: the free boundary flat $120^\circ$88-cone is a singular Plateau model surface with three planar faces meeting at $120^\circ$89, rigid under conformal free-boundary minimal immersion and characterized by Morse index $120^\circ$90 in $120^\circ$91 (Matinpour, 29 Sep 2025). Many nearby arXiv literatures discuss “free boundary cones,” but these are usually rotational Bernoulli cones, capillary hypersurface cones, or wedge-hyperplane sections rather than genuine $120^\circ$92-junction configurations (Matinpour, 26 May 2026).