Disc bounded by a plane curve Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.

Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?

It is easy to find an open disc which boundary lies in a plane,
but the boundary might be crazy; for example it might be Polish circle shown on the diagram.

Comments

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*A simpler question: Does every surface topologically embedded in the Euclidean space contains a planar arc?


*Curves on potatoes --- another closely related problem.
 A: As requested, here is a related result:
Theorem. There exists a continuous (also also open) real-valued function $R^2\to R$ whose level sets are all homeomorphic to the open pseudo-arc. In particular, level sets contain no nondegenerate arcs.
Similarly, one can construct a continuous monotone map $S^2\to [0,1]$ such that all level sets are homeomorphic to the pseudo-circle, i.e. the unique (up to homeomorphism) hereditarily indecomposable circularly chainable continuum. Again, pseudo-circle contains no nondegenerate arcs.
The story of this theorem is rather interesting. Quoting from
Prajs, Janusz R., A continuous circle of pseudo-arcs filling up the annulus, Trans. Am. Math. Soc. 352, No. 4, 1743-1757 (2000). ZBL0936.54019.
where a proof of this theorem is given:

Among the results obtained by Knaster during World War II one can find the following announcement, originally presented in Kiev in 1940:


There exists a real-valued, monotone mapping from the plane that is not constant on any arc. In the construction Knaster’s hereditarily indecomposable continua were exploited. Actually, Knaster’s result can be reformulated in the following stronger version:
There exists a real valued, monotone mapping from the plane such that all pointinverses are hereditarily indecomposable.


Unfortunately, Knaster’s notes concerning this result were burned during the war, Knaster had never written down the result again, and even his closest exstudents do not know his original idea of construction...


A result similar to that announced by Knaster (with higher dimensional analogues) was proved by Brown in 1958:


M. Brown, Continuous collections of higher dimensional continua, Ph. D. Thesis, University
of Wisconsin, 1958.

Of course, Brown did not publish his proof either, but Prajs finally did, see above.
