A submanifold M in a Euclidean space is said to be of finite type if its position vector field is a finite sum of (vectorial) eigenfunctions of the Laplace operator Δ of M.
For surfaces in E3, this conjecture is true for tubes, ruled surfaces, quadrics, sprial surfaces, surfaces of Dupin, translation surfaces, and rotational surfaces (see pp. 179-188 of my 2015 book [Total mean curvature and submanifolds of finite type, 2nd edition, World Scientific, 2015]).
A Euclidean submanifold is said to be of null 2-type if its position vector field x is the sum x = x0 + x1 where ∆x0 = 0 and ∆x1 = λx1, where λ is a nonzero real number and x0 is a non-constant map.
My conjecture related to this problem is that the answer to this open problem is affirmative.
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