Critical Catenoid: Geometry & Spectrum

• Critical catenoid is a free-boundary minimal annulus obtained by truncating and scaling the standard catenoid to meet the sphere orthogonally.
• Its spectral structure, analyzed via Jacobi operators and Robin conditions, reveals a Morse index of 4 and a nullity of 2.
• It serves as a rigidity target, benchmark for numerical methods, and foundational building block in constructing more complex minimal surfaces.

Searching arXiv for recent and foundational papers on the critical catenoid. arXiv search query: "critical catenoid free boundary minimal surface Morse index" The critical catenoid is the rotationally symmetric free-boundary minimal annulus in the Euclidean unit ball $B^3\subset \mathbb{R}^3$ obtained by truncating and scaling the standard catenoid so that it meets the boundary sphere $S^2=\partial B^3$ orthogonally. In the modern literature it functions simultaneously as a model example in free-boundary minimal surface theory, a rigidity target in uniqueness theorems, and a benchmark for spectral and numerical methods. Its classical Euclidean form has Morse index $4$, and recent spectral treatments record nullity $2$ (Devyver, 2016, Smith et al., 2016, Samarakkody, 19 Jun 2026).

1. Geometric definition and explicit parametrization

A free-boundary minimal surface $\Sigma\subset B^3$ is a compact surface with $\partial\Sigma\subset S^2$ such that $\Sigma$ has zero mean curvature in the interior and meets $S^2$ orthogonally along $\partial\Sigma$ (Devyver, 2016, McGrath, 2016). For the critical catenoid, one starts from the standard catenoid in $\mathbb{R}^3$ and imposes the free-boundary condition.

One convenient parametrization is

$S^2=\partial B^3$0

with a unique parameter choice for which the truncated surface

$S^2=\partial B^3$1

meets the unit sphere orthogonally. Here $S^2=\partial B^3$2 is the unique solution of

$S^2=\partial B^3$3

and then

$S^2=\partial B^3$4

(Devyver, 2016). An equivalent scaled parametrization writes

$S^2=\partial B^3$5

where $S^2=\partial B^3$6 is the unique solution of

$S^2=\partial B^3$7

(McGrath, 2016). The equivalent forms

$S^2=\partial B^3$8

also appear in the literature (Smith et al., 2016, Ambrozio et al., 2016).

Topologically, the critical catenoid is an annulus with two boundary circles, and geometrically it is invariant under rotations about its axis and reflection across the equatorial plane (Kapouleas et al., 2017). The term “critical” refers to the fact that only one scale and truncation of the classical catenoid produces a compact piece lying in the unit ball and meeting $S^2=\partial B^3$9 orthogonally (Kapouleas et al., 2017).

2. Variational formulation and Jacobi spectrum

For a free-boundary minimal surface in the unit ball, the second variation of area is governed by the quadratic form

$4$0

with Jacobi operator

$4$1

and Robin boundary condition

$4$2

(Devyver, 2016, Smith et al., 2016). This is the free-boundary stability problem.

For the critical catenoid, conformal coordinates simplify the analysis. In the coordinates $4$3, the induced metric is

$4$4

and after a conformal change one obtains a spherical annulus model on which the Jacobi operator becomes

$4$5

(Devyver, 2016). Separation of variables then reduces the spectral problem to one-dimensional ODEs indexed by Fourier mode.

A Jacobi-Steklov formulation makes this reduction especially explicit. With

$4$6

one obtains the ODE

$4$7

with Robin conditions

$4$8

where $4$9 is the Jacobi-Steklov eigenvalue (Samarakkody, 19 Jun 2026).

This spectral structure admits several equivalent interpretations. In one description, the constant function together with three translation Jacobi fields span a four-dimensional negative subspace for the second variation (Devyver, 2016). In another, the index and nullity are assembled from the one-dimensional Robin problems and the fixed-boundary Dirichlet spectrum (Samarakkody, 19 Jun 2026). These formulations are complementary rather than contradictory: they encode the same instability through different decompositions.

3. Morse index, nullity, and mode-by-mode analysis

The Euclidean critical catenoid has Morse index $2$0 (Devyver, 2016, Smith et al., 2016). One proof proceeds by identifying four explicit negative directions—one constant mode and three translation Jacobi fields—and then showing that all orthogonal higher modes are nonnegative (Devyver, 2016). Another proof separates variables and analyzes the resulting ODEs by Sturm-Liouville and Riccati-type arguments (Smith et al., 2016).

In the mode decomposition of (Smith et al., 2016), the $2$1 sector contributes two negative eigenvalues, one even and one odd, while the $2$2 sector contributes one negative eigenvalue of multiplicity two; no negative eigenvalues occur for $2$3 (Smith et al., 2016). This yields

$2$4

negative directions in total.

A Jacobi-Steklov computation gives a finer accounting. The only eigenvalues strictly below the threshold $2$5 are

$2$6

with multiplicity $2$7, and

$2$8

with multiplicity $2$9. The next eigenvalue is $\Sigma\subset B^3$0 with multiplicity $\Sigma\subset B^3$1, and the Dirichlet problem has one zero eigenvalue and no negative eigenvalues. Consequently,

$\Sigma\subset B^3$2

(Samarakkody, 19 Jun 2026).

This spectral picture is also consistent with a general lower bound: every non-flat free-boundary minimal surface in $\Sigma\subset B^3$3 has index at least $\Sigma\subset B^3$4, while the flat equatorial disk is the exceptional index-$\Sigma\subset B^3$5 case (Devyver, 2016). The critical catenoid therefore realizes the smallest Morse index among non-flat examples in the Euclidean three-ball, as recorded in that framework (Devyver, 2016).

4. Characterizations and rigidity results in the unit ball

The critical catenoid is the conclusion of several distinct rigidity theorems. These results use different hypotheses—index, Steklov spectrum, symmetry, support-function structure, or curvature pinching—but each singles out the same annulus.

Hypothesis	Conclusion	Source
Embedded free-boundary minimal surface in $\Sigma\subset B^3$6 with index $\Sigma\subset B^3$7	It is the critical catenoid	(Espinar et al., 2016)
Embedded free-boundary minimal annulus invariant under the three coordinate-plane reflections	It is the critical catenoid	(McGrath, 2016)
Embedded minimal free-boundary annulus whose boundary is invariant under the three coordinate-plane reflections	It is the critical catenoid	(Barbosa et al., 2020)
Embedded free-boundary minimal annulus with two congruent boundary components and one reflection symmetry	It is congruent to the critical catenoid	(Seo, 2021)
Embedded free-boundary minimal annulus whose support function has infinitely many critical points	It is the critical catenoid	(Espinar et al., 2023)
Pinching $\Sigma\subset B^3$8, with equality somewhere	It is congruent to the critical catenoid	(Ambrozio et al., 2016, Barbosa et al., 2018)

A central spectral characterization comes from the Steklov problem. For a free-boundary minimal annulus in $\Sigma\subset B^3$9, the coordinate functions restrict to Steklov eigenfunctions with eigenvalue $\partial\Sigma\subset S^2$0, and Fraser–Schoen’s theorem identifies the critical catenoid as the unique free-boundary minimal annulus whose coordinate functions realize the first Steklov eigenvalue (McGrath, 2016). McGrath’s reflection-symmetry theorem then shows that coordinate-plane invariance forces $\partial\Sigma\subset S^2$1, hence forces the critical catenoid (McGrath, 2016).

A PDE-based characterization due to Espinar–Marín starts from the support function

$\partial\Sigma\subset S^2$2

pushes it through the Gauss map, and obtains a function $\partial\Sigma\subset S^2$3 on a spherical domain $\partial\Sigma\subset S^2$4 satisfying

$\partial\Sigma\subset S^2$5

Under the additional assumption that $\partial\Sigma\subset S^2$6 has infinitely many maximum points, the solution is rotationally symmetric, and the corresponding embedded free-boundary minimal annulus is the critical catenoid (Espinar et al., 2023).

A different rigidity mechanism is curvature pinching. Ambrozio–Nunes proved that if a compact free-boundary minimal surface in $\partial\Sigma\subset S^2$7 satisfies

$\partial\Sigma\subset S^2$8

everywhere, then either the quantity vanishes identically and the surface is the flat equatorial disk, or equality occurs at some point and the surface is congruent to the critical catenoid (Ambrozio et al., 2016). Barbosa–Viana extended the same dichotomy to arbitrary codimension, showing that equality still forces the surface to lie in a three-dimensional linear subspace and be the critical catenoid (Barbosa et al., 2018).

Taken together, these theorems provide several independent rigidity routes to the same object. In the cited literature, the critical catenoid is therefore characterized not by a single universal criterion, but by a network of spectral, geometric, and variational signatures.

5. Higher-dimensional and hyperbolic analogues

The term “critical catenoid” also appears in higher-dimensional free-boundary theory. For every $\partial\Sigma\subset S^2$9, the $\Sigma$0-dimensional critical catenoid $\Sigma$1 is defined as the unique nontrivial-topology, rotationally symmetric, free-boundary minimal hypersurface embedded in the closed unit ball $\Sigma$2 (Smith et al., 2017). Its Morse index $\Sigma$3 obeys the asymptotic formula

$\Sigma$4

(Smith et al., 2017). The same work computes exact values for $\Sigma$5, beginning with

$\Sigma$6

(Smith et al., 2017). In this sense, the classical surface in $\Sigma$7 is the $\Sigma$8 member of a broader unstable family.

A distinct analogue arises in hyperbolic space. In $\Sigma$9, the Mori family provides rotationally symmetric minimal immersions $S^2$0, and for each $S^2$1 one obtains a critical hyperbolic catenoid

$S^2$2

that is free-boundary minimal in a geodesic ball (Pigazzini, 13 May 2026). The Medvedev conjecture states

$S^2$3

and its strong form asserts

$S^2$4

(Pigazzini, 13 May 2026).

The 2026 local resolution proves that there exists $S^2$5 such that for every

$S^2$6

one has

$S^2$7

(Pigazzini, 13 May 2026). The proof reduces the problem to one-dimensional spectral inequalities and to the monotonicity of

$S^2$8

using a real-analytic expansion of $S^2$9 near $\partial\Sigma$0 with positive linear coefficient $\partial\Sigma$1 (Pigazzini, 13 May 2026). This places the hyperbolic theory in direct conceptual continuity with the Euclidean index problem, but the two settings remain analytically distinct.

6. Role as a model surface in construction and computation

The critical catenoid is not only a rigidity target; it is also a building block for new free-boundary minimal surfaces. Kapouleas–Li construct a family of high-genus examples in the Euclidean unit three-ball by desingularizing the intersection of a coaxial pair consisting of a critical catenoid and an equatorial disk (Kapouleas et al., 2017). The resulting surfaces have three boundary components and are described as the free-boundary analogue of the Costa–Hoffman–Meeks surfaces and of Kapouleas’s constructions obtained by desingularizing coaxial catenoids and planes (Kapouleas et al., 2017). This places the critical catenoid at the base of a gluing theory for more complicated topologies.

It also serves as a stringent numerical benchmark. A physics-informed neural network study uses the Jacobi-Steklov reduction to one-dimensional Robin problems, enforces parity mode by mode, and treats the eigenvalue as a trainable parameter (Samarakkody, 19 Jun 2026). In that setting, the three eigenvalues below the stability threshold are recovered to within $\partial\Sigma$2 to $\partial\Sigma$3 of their exact values, with PDE residuals of order $\partial\Sigma$4, and the assembled spectrum reproduces

$\partial\Sigma$5

(Samarakkody, 19 Jun 2026).

The same work studies an operator homotopy joining a flat reference operator to the catenoid Jacobi operator. Numerically, the index is $\partial\Sigma$6 at the flat strip endpoint, rises to $\partial\Sigma$7 for small positive homotopy parameter, returns to $\partial\Sigma$8 for an intermediate range, and equals $\partial\Sigma$9 at the catenoid endpoint (Samarakkody, 19 Jun 2026). Because the critical catenoid is rigid in that analysis, the homotopy deforms operators rather than surfaces (Samarakkody, 19 Jun 2026). This suggests a broader methodological role: the critical catenoid is especially useful when one wants a problem whose geometry, exact spectrum, and instability count are all known in closed form, but whose analytic structure remains rich enough to test new techniques.