Geometry of Curves and Surfaces — Problem 1.3

Definitions and mathematical statement

Let $A$ be a compact smooth annulus with a Riemannian metric $g$ of strictly negative Gauss curvature in its interior. Prescribe two disjoint smooth simple closed convex planar curves $\Gamma_0,\Gamma_1\subset\mathbb R^3$ as its boundary components. Each curve lies in a plane; the planes need not be the same. Consider smooth embeddings $f:A\to\mathbb R^3$ inducing the metric $g$ and carrying the two boundary circles onto these fixed curves.

The rigidity question asks whether two such isometric realizations with the same boundary data must be congruent by an ambient Euclidean isometry respecting that data. In a marked formulation, fix the boundary parametrizations as well and compare embeddings that agree there. In particular, rigidity would prohibit a nontrivial continuous isometric deformation with the boundary held fixed.

The negative curvature concerns the intrinsic metric of the annulus, equivalently the product of its two principal curvatures under an isometric embedding. Fixing the boundary means preserving the actual space curves, not only their lengths or their planar congruence classes. The problem is about intrinsic isometries of surfaces; having the same Gauss curvature function alone is a weaker requirement and is not the proposed rigidity hypothesis.

Short English statement

Is a negatively curved annulus uniquely determined, up to rigid motion, by its intrinsic metric and its two fixed convex planar boundary curves?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000003 (AMR-069-0003), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ghomi - Open Problems in Geometry of Curves and Surfaces (2019), Problem 1.3, PDF page 6. Original source indexed by AMR-069-0003. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.