Free Boundary Plateau Model Cones in $\mathbb{B}^n$ are Rigid under Conformal Minimal Immersions

Abstract: The classical theorem of Nitsche asserts that every free-boundary minimal disk in the unit ball $\mathbb{B}<sup>3$ is an equatorial flat disk. Fraser and Schoen later generalized this rigidity theorem to arbitrary dimensions and ambient spaces of constant sectional curvature. In previous work, the author established an analogous rigidity result for the singular $Y$-cone: any conformal free-boundary minimal immersion of the flat $Y$-cone into $\mathbb{B}<sup>n$ is congruent to the flat $Y$-cone. In this paper we treat the remaining classical two-dimensional Plateau singularity model, namely the tetrahedral $T$-cone. We prove that every conformal free-boundary minimal immersion of the flat $T$-cone into $\mathbb{B}<sup>n$ is congruent, up to an orthogonal transformation, to the flat $T$-cone itself. As a consequence, combining this result with the Nitsche--Fraser--Schoen theorem and the previously established $Y$-cone rigidity theorem, we obtain a unified rigidity theorem for the classical Plateau model domains: any free-boundary minimal Plateau surface in $\mathbb{B}<sup>n$ conformal to a plane disk, a $Y$-cone, or a $T$-cone must be congruent to the corresponding model.

• The paper establishes that every conformal free-boundary minimal immersion of the flat tetrahedral T-cone into the unit ball coincides with it up to an orthogonal transformation.
• Its proof combines a Hopf differential argument, spherical network rigidity, and reflection principles to show that boundary arcs are geodesic and all six faces are planar.
• Together with earlier disk and Y-cone results, the theorem gives a unified conformal rigidity result for the three classical two-dimensional Plateau model cones, while leaving the non-conformal case open.

This paper by Elham Matinpour establishes a rigidity theorem for free-boundary minimal surfaces with tetrahedral ($T$-type) Plateau singularities in the unit ball $\mathbb{B}^n$. The main result states that any conformal minimal immersion of the flat $T$-cone into $\mathbb{B}^n$ meeting the boundary sphere orthogonally must coincide, up to an orthogonal transformation, with the flat $T$-cone itself. Combined with the classical Nitsche–Fraser–Schoen rigidity of free-boundary minimal disks and the author's earlier $Y$-cone result (Matinpour, 29 Sep 2025), this yields a unified uniqueness theorem covering all three classical two-dimensional Plateau model domains: the disk, the $Y$-cone, and the $T$-cone.

Background and context

The starting point is Nitsche's theorem that every free-boundary minimal disk in $\mathbb{B}^3$ is an equatorial flat disk [nitsche1985stationary], later extended by Fraser and Schoen to all dimensions and to space forms, where such disks are totally geodesic [fraser2015uniqueness]. The singular analogue concerns Plateau surfaces—area-stationary integral currents whose supports are locally modeled on the plane $P$, half-plane $\mathbb{B}^n$0, the $\mathbb{B}^n$1-cone (three half-planes at $\mathbb{B}^n$2), or the $\mathbb{B}^n$3-cone (the cone over the 1-skeleton of a regular tetrahedron). A minimal $\mathbb{B}^n$4-surface consists of six faces meeting along four junction curves in triples at $\mathbb{B}^n$5, with a single $\mathbb{B}^n$6-vertex where the local configuration matches the cone over the tetrahedral skeleton. The free-boundary condition requires each face to be minimally immersed in $\mathbb{B}^n$7, the boundary to lie on $\mathbb{B}^n$8, orthogonality against $\mathbb{B}^n$9, and conormal balance $T$0 along junctions.

The paper's conformality assumption is a genuine restriction: the map $T$1 must be conformal on each face with respect to the flat metric induced from the model cone, agree across junctions, and have matching tangential differentials. Whether the rigidity theorem holds without conformality is not addressed here and remains open.

Rigidity of tetrahedral stationary geodesic networks

The first step is a self-contained lemma showing that any stationary geodesic network on $T$2 with tetrahedral combinatorics—four vertices, six geodesic edges, every pair connected—is congruent to the regular tetrahedral network. The proof proceeds in four stages: the $T$3 equilibrium condition forces the tangent directions at each vertex to span a 2-plane, so all vertices lie in a common 3-dimensional subspace; Euler's formula gives exactly four spherical triangular regions; the spherical law of sines shows each triangle is equilateral since its angles are all $T$4; and the spherical law of cosines yields $T$5, hence edge length $T$6, identifying the configuration as a regular tetrahedron. Degenerate cases are excluded by the embedding and finiteness assumptions. This lemma reduces the boundary problem to a fixed spherical link, independent of the ambient dimension.

Free-boundary minimal $T$7-surfaces spanning the regular network

The second main step proves that any free-boundary minimal $T$8-surface whose spherical boundary is the regular tetrahedral network must be the flat $T$9-cone. For each face $\mathbb{B}^n$0 bounded by a great-circle arc $\mathbb{B}^n$1 and two junction curves, the reflection principle for free-boundary minimal surfaces (attributed to Choe [choe2025free]) reflects $\mathbb{B}^n$2 across $\mathbb{B}^n$3 along $\mathbb{B}^n$4, producing a real-analytic minimal surface $\mathbb{B}^n$5 for which $\mathbb{B}^n$6 is interior. Because the plane $\mathbb{B}^n$7 through the origin containing $\mathbb{B}^n$8 is invariant under the reflection, and because the free-boundary condition makes the position vector tangent to $\mathbb{B}^n$9 along $T$0, the tangent planes of $T$1 and $T$2 coincide along $T$3. The squared distance function to $T$4 is subharmonic on the minimal surface, vanishes to first order along the interior curve $T$5, and therefore vanishes identically by unique continuation; consequently $T$6. Since this holds for all six faces, planarity forces each junction to be a straight segment from the $T$7-point to a boundary vertex, and the orthogonal-meeting condition places the $T$8-vertex at the origin. An alternative version of the argument invokes Lawlor's theorem for straightness of junctions and a monotonicity-formula argument to force the vertex to the origin.

The Hopf differential argument

The final ingredient removes the a priori assumption that the boundary network is geodesic. On each quarter-disk sector $T$9, the normal part $Y$0 of the complex second derivative satisfies the classical identity $Y$1, so it is holomorphic. Summing over the six faces gives a holomorphic quadratic differential $Y$2, and setting $Y$3, the key computation shows:

• On the free-boundary arc $Y$4: the condition $Y$5 implies $Y$6, making $Y$7 real-valued.
• On the junction arcs $Y$8: matching of first and second radial derivatives across the triple junctions, together with harmonicity and the $Y$9-balance condition $Y$0, forces $Y$1.

Thus $Y$2 is holomorphic on $Y$3 and real on the entire boundary, hence constant; since $Y$4, it vanishes identically. Reading off real and imaginary parts gives $Y$5 and $Y$6 along $Y$7, and minimality then forces both normal second derivatives to vanish. The second fundamental form of each face therefore vanishes along its spherical boundary arc, which means each arc is a great-circle piece: the boundary is a stationary geodesic network with tetrahedral combinatorics, and the previous section's theorem applies. This Hopf differential technique is adapted directly from the author's $Y$8-cone proof (Matinpour, 29 Sep 2025), with the tetrahedral combinatorics supplying the necessary cancellation along the junctions.

Limitations and open questions

The rigidity theorem is proved only under the conformality hypothesis on the immersion relative to the flat model metric; the paper does not claim uniqueness among arbitrary (non-conformal) free-boundary minimal $Y$9-surfaces, and the question of whether conformality can be dropped is left open. Similarly, the analysis is restricted to $T$0-surfaces with exactly one $T$1-point, six faces, and four junctions—the single-tetrahedron topology—and does not treat configurations with multiple $T$2-points or higher-genus Plateau complexes. The reflection-principle input relies on Choe's recent work [choe2025free], so the result inherits whatever regularity hypotheses that principle requires near the boundary. Finally, the unified theorem covers only the three classical model cones; whether analogous rigidity holds for other stationary cone types or in ambient manifolds of nonconstant curvature is not addressed.

Conclusion

The paper completes a program initiated with the $T$3-cone: under the conformality assumption, each of the flat disk, $T$4-cone, and $T$5-cone is rigid among free-boundary minimal Plateau surfaces in $T$6 modeled on the corresponding singularity type. The proof combines a geometric uniqueness lemma for spherical Steiner networks, a reflection-and-maximum-principle argument forcing facial planarity, and a summed-Hopf-differential computation showing the boundary network is geodesic. The natural next questions raised by the work concern removing the conformality assumption and extending the uniqueness theory beyond the single-$T$7-point topology.