Finiteness of embedded minimal surfaces in S^3 of fixed genus

Determine whether, for every genus γ ≥ 2, the set M_γ of closed embedded minimal surfaces in S^3 of genus γ, modulo isometries of S^3, is finite.

Background

The paper surveys known results on embedded minimal surfaces in S^3 and recalls that, while existence and compactness are established for each genus, the exact count of distinct embedded minimal surfaces up to isometry is not known.

The authors emphasize that although many families have been constructed and lower bounds on counts are known, deciding finiteness for each fixed genus remains unresolved.