Y-catenoid: Singular Minimal Surface Model

• Y-catenoid is a singular minimal surface with a Y-point junction, comprising a planar disk and two catenoidal sheets meeting at 120° angles.
• It serves as the canonical Morse index one example in the theory of Y-singular minimal surfaces, with rotational symmetry simplifying its analysis.
• Its index analysis, involving fixed boundary and Steklov eigenvalue problems, clarifies stability properties distinct from classical smooth catenoids.

The $Y$-catenoid is a rotationally symmetric $Y$-singular minimal surface in $\mathbb{R}^3$ whose singular set is a single circle and whose geometry combines one planar disk with two symmetric catenoidal faces meeting along a common junction at $120^\circ$. In the terminology used for $Y$-singular Plateau-type surfaces, it is a model example of a surface with a $Y$-point singularity pattern, and recent work identifies it as the canonical Morse index one example in the low-index theory of minimal surfaces with $Y$-singularities (Matinpour, 2024).

1. Definition and geometric model

A $Y$-singular minimal surface is a minimal surface that is allowed to have singular junctions of $Y$-type rather than being smooth everywhere. In this framework, a $Y$-point is a singular point where the tangent configuration consists of three sheets meeting at equal $Y$0 angles, like the standard $Y$1-cone. A $Y$2-surface is a Plateau-type minimal surface with only $Y$3-singularities and no $Y$4-points. A convenient model is a triple junction surface, consisting of three smooth minimal pieces $Y$5 glued along a common curve $Y$6, with compatibility conditions ensuring that the three sheets meet in the $Y$7-pattern (Matinpour, 2024).

For the $Y$8-catenoid, the simplifying assumptions are rotational symmetry and the condition that the $Y$9-singularity set is one circle

$\mathbb{R}^3$0

centered at the origin. The surface is written as

$\mathbb{R}^3$1

where $\mathbb{R}^3$2 is the disk bounded by $\mathbb{R}^3$3, and $\mathbb{R}^3$4 are two catenoidal sheets meeting $\mathbb{R}^3$5 along $\mathbb{R}^3$6 at the required $\mathbb{R}^3$7 angle condition. Geometrically, it is the limiting $\mathbb{R}^3$8-noid configuration as the parameter $\mathbb{R}^3$9, with the horizontal plane removed (Matinpour, 2024).

This description distinguishes the $120^\circ$0-catenoid from the classical smooth catenoid. The latter is a smooth minimal annulus, whereas the $120^\circ$1-catenoid is singular along a compact interface and decomposes into three faces meeting along that interface.

2. Variational structure and admissible deformations

The analytic study of the $120^\circ$2-catenoid is organized through the stability operator on each face,

$120^\circ$3

where $120^\circ$4 is the Laplace–Beltrami operator and $120^\circ$5 is the squared norm of the second fundamental form. For a triple junction variation $120^\circ$6, the second variation is encoded by

$120^\circ$7

The Morse index is the maximal dimension of a subspace of admissible variations on which $120^\circ$8 is negative definite. Equivalently, it counts the number of independent directions in which the surface is unstable (Matinpour, 2024).

A related decomposition is

$120^\circ$9

This separates interior Jacobi contributions from junction terms and is central in the index calculation.

In the broader low-index theory, admissible normal variations must respect the $Y$0-junction. For a normal variation $Y$1 on the faces $Y$2, the compatibility condition is

$Y$3

The corresponding Sobolev space is

$Y$4

and the second variation quadratic form extends to

$Y$5

This is the natural framework for discussing complete two-sided minimal $Y$6-surfaces and their Morse index (Matinpour, 29 Sep 2025).

3. Computation of the Morse index

The index computation for the $Y$7-catenoid is reduced to two auxiliary problems. The first is the fixed boundary problem on the singularities, which counts variations that do not move the interface $Y$8. These are functions satisfying

$Y$9

The relevant quantity is the Dirichlet index of each face,

$Y$0

For the $Y$1-catenoid, each face is stable under such fixed-boundary variations, so

$Y$2

Thus there are no unstable directions coming from deformations that keep the junction circle fixed (Matinpour, 2024).

The second ingredient is a Dirichlet-to-Neumann / Steklov problem for the stability operator, which treats variations that move the interface. A function $Y$3 on $Y$4 is a $Y$5-Steklov eigenfunction if

$Y$6

Equivalently, with $Y$7,

$Y$8

The boundary contribution is packaged into a quadratic form on coefficient vectors $Y$9 satisfying

$Y$0

namely

$Y$1

Because the $Y$2-catenoid is rotationally symmetric, the kernel of the Jacobi operator on a catenoidal face can be written explicitly by separation of variables. The relevant bounded $Y$3-modes on the catenoidal faces are: $Y$4 and

$Y$5

while higher Fourier modes $Y$6 are stable in the index sense. For the planar face, the Jacobi operator is the Laplacian, and the bounded modes are standard harmonic functions. After evaluating the boundary terms $Y$7, one linear combination of the $Y$8-type modes contributes a negative direction, the $Y$9-type modes contribute to the nullity, and all higher modes are nonnegative (Matinpour, 2024).

A concise summary is given below.

Component of the index analysis	Conclusion for the $Y$0-catenoid
Fixed boundary problem	$Y$1
Lowest radial mode $Y$2	Produces one negative direction
Angular modes $Y$3	Contribute to the nullity
Higher Fourier modes $Y$4	Nonnegative

4. Index one and nullity three

The main theorem of the index computation is

$Y$5

The same analysis states that the nullity is $Y$6. Interpreted variationally, index $Y$7 means that there is exactly one independent admissible deformation that decreases area to second order, while all other infinitesimal deformations are neutral or stabilizing (Matinpour, 2024).

This result places the $Y$8-catenoid at the threshold between stability and higher-mode instability. The surface is not stable, but its instability is minimal in the Morse-theoretic sense. A plausible implication is that it occupies, within the singular category, the same structural niche that the classical catenoid occupies among smooth complete minimal surfaces of low index.

That interpretation is made explicit in later rigidity work, which recalls the smooth hierarchy and then places the singular case alongside it: index $Y$9 corresponds to rigidity in the stable case, index $Y$0 is the first nontrivial unstable regime, and the $Y$1-catenoid is the distinguished singular model realizing that regime. The same work notes that the $Y$2-catenoid has Morse index one, whereas a family of $Y$3-noids has index two (Matinpour, 29 Sep 2025).

5. Rigidity and classification in the low-index theory

The $Y$4-catenoid acquires a broader structural role in the classification of complete, two-sided minimal $Y$5-surfaces in $Y$6. In that setting, the surface is treated as a rectifiable current mod 3, complete and without boundary, two-sided, minimal, and with Euclidean area growth. The allowed local models are a plane, a half-plane, or a $Y$7-cone, where a $Y$8-cone is a union of three half-planes joined along their common boundary line, meeting at $Y$9 angles (Matinpour, 29 Sep 2025).

A main theorem in this setting states that if $Y$0 is a complete, two-sided, minimal $Y$1-surface in $Y$2 with three faces $Y$3, compact interface $Y$4, and Morse index one, then up to relabeling:

• $Y$5 is a compact disk with total curvature at most $Y$6,
• $Y$7 is an unbounded annulus with an embedded end and total curvature at most $Y$8,
• $Y$9 is unbounded and (Dirichlet) stable.

In addition, if $Y$00 has only one end then $Y$01 is a $Y$02-catenoid (Matinpour, 29 Sep 2025).

Within this theorem, the $Y$03-catenoid is the distinguished index-one example among complete, two-sided minimal $Y$04-surfaces. The proof uses the weighted space

$Y$05

constant compatible test functions

$Y$06

and a cutoff argument yielding the approximate expression

$Y$07

The sign constraints imposed by index one force one face to be a compact disk, one an unbounded annulus, and one unbounded and stable. The $Y$08-catenoid is then singled out under the additional one-end hypothesis on the stable face.

6. Related objects, terminology, and potential ambiguities

The term $Y$09-catenoid should not be conflated with other catenoid-type constructions. A related but distinct object is the “vase of catenoids,” a symmetric immersed minimal surface on a punctured sphere with one planar end and multiple catenoid ends. That construction is minimal, immersed, symmetric, and built using Weierstrass data, but the source explicitly states that it does not define or name a “$Y$10-catenoid” (Connor, 2016).

Likewise, work on catenoid stability with one pinned and one free contact line analyzes a zero-mean-curvature catenoid bridge with a free boundary on a substrate, but the term “$Y$11-catenoid” does not appear there either. That problem concerns liquid bridges and perturbative stability rather than triple-junction minimal surfaces with $Y$12-singularities (Akbari et al., 2015).

The defining feature of the $Y$13-catenoid is therefore not merely the presence of catenoidal geometry, symmetry, or multiple ends. It is specifically the triple-junction singular structure: three faces meeting along a compact interface in the $Y$14-pattern, together with the Morse-theoretic property that the resulting minimal surface has index one and nullity three in the rotationally symmetric single-circle case (Matinpour, 2024).

In that sense, the $Y$15-catenoid functions as the canonical low-index model in the singular theory. The available results suggest a parallel with the role of the classical catenoid in the smooth category: it is the simplest nontrivial unstable object, and rigidity theorems identify it as the distinguished geometry once the admissible topology and the Morse index are sufficiently constrained (Matinpour, 29 Sep 2025).