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Spherical cap conjecture

Let X:D‾→R3X:\overline{\mathbb{D}}\to\mathbb{R}^{3} be a smooth immersion of a disk, regular up to the boundary, with nonzero constant mean curvature. If X∣∂DX|_{\partial\mathbb{D}} maps ∂D\partial\mathbb{D} diffeomorphically onto a circle, then X(D‾)X(\overline{\mathbb{D}}) is an embedded spherical cap.

References

Primary source

arXiv

Additional references

differential-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle the smooth immersed-disk case in three-dimensional Euclidean space, while broader versions remain open.

The spherical cap conjecture asserts that a compact constant-mean-curvature surface with circular boundary must be a spherical cap under specified genus, immersion, or embedding assumptions. A 2025 survey stated that the general conjecture remained open; no original proposer is identified in the retrieved sources.

Known results

  • Alías, López, and Palmer: stable immersed constant-mean-curvature discs bounded by a circle are umbilical.
  • Koyso: a half-space criterion under an exterior-intersection restriction.
  • Brito, Sá Earp, Meeks, and Rosenberg: one-sidedness under convexity and transversality assumptions.
  • Cruz and Nelli, 2025: higher-dimensional and embedded results under local-convexity hypotheses, not the full conjecture.

September 2026 immersed-disk result

A September 21, 2026 news entry reports that José M. Espinar’s preprint, The spherical cap conjecture for immersed disks, establishes the stated classification in Euclidean three-space. This claims to rule out non-spherical examples under the precise smooth-disk and circular-boundary hypotheses, but the claim is unverified here.

Current status (as of September 2026): The precise smooth immersed-disk case in R3\mathbb{R}^3 is claimed solved, while broader embedded or higher-dimensional formulations and independent verification remain open.

Sources

Solutions 0

No solutions have been posted yet.