Geometry of Curves and Surfaces — Problem 1.5

Definitions and mathematical statement

Let $D$ be a compact smooth two-dimensional disk with a smooth Riemannian metric $g$ of positive Gauss curvature. Let $\Gamma\subset\mathbb R^3$ be a smooth embedded closed curve. An isometric spanning disk is a smooth embedding $f:D\to\mathbb R^3$ satisfying \[ f^*\langle\ ,\ \rangle=g, \qquad f(\partial D)=\Gamma. \] The pullback condition means that for all tangent vectors $v,w$ at every point, $\langle df(v),df(w)\rangle=g(v,w)$. No boundary parametrization is prescribed in the unmarked version; a marked version supplies one and requires $f$ to extend it.

Find necessary and sufficient conditions on $g$ and $\Gamma$ for such an embedding to exist. Conditions must describe the given data and be more informative than restating that a solution exists. Matching the curve length with the $g$-length of $\partial D$ is necessary, but is not by itself the requested complete criterion.

The entire intrinsic metric is prescribed, not merely a positive curvature sign or a total area. The disk must be embedded, so an immersed spanning surface with self-intersections does not solve the problem. Its boundary is the specified space curve, which need not be planar.

Short English statement

Characterize which space curves can bound a positively curved disk with a specified intrinsic metric, without stretching or self-intersection.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000005 (AMR-069-0005), 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 1.5, PDF page 7. Original source indexed by AMR-069-0005. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.