Geometry of Curves and Surfaces — Problem 6.3

Definitions and mathematical statement

Let $M\subset\mathbb R^3$ be a smooth compact connected embedded orientable surface with boundary. Choose a continuous unit normal and define its mean curvature by $H=(k_1+k_2)/2$, where $k_1,k_2$ are the corresponding principal curvatures. Assume $H$ is constant throughout the interior.

Suppose $\partial M$ is one smooth simple closed convex curve $\Gamma$ in an affine plane $\Pi$, and the entire surface lies in one of the closed half-spaces bounded by $\Pi$. Is $M$ necessarily homeomorphic to a closed disk?

The assertion is topological: it rules out handles on a one-boundary-component surface. It does not claim that the surface is planar, spherical, or uniquely determined by $\Gamma$. The curve being convex means that it bounds a convex planar region; convexity of the whole surface is not assumed. The half-space may be either side of the boundary plane, but it is fixed for the entire surface.

Embeddedness excludes self-intersections, and no stability or area-minimizing hypothesis is imposed. Adding such a variational assumption gives a special case rather than the full question. Reversing the chosen normal changes the sign of $H$ but does not change the constant-mean-curvature condition or the problem.

Short English statement

Must an embedded constant-mean-curvature surface with one convex planar boundary curve have no handles if it stays on one side of the boundary plane?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000025 (AMR-069-0025), 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 6.3, PDF page 14. Original source indexed by AMR-069-0025. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.