Vol. 30, No. 3, 1969

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Holomorphic quadratic differentials on surfaces in E3

Tilla Weinstein

Vol. 30 (1969), No. 3, 697–715

Let R be a Riemann surface defined upon an oriented surface S smoothly immersed in E3. This paper studies holomorphic quadratic differentials on R which are related to the geometry on S, especially those of the form

Ωˆ = {(A − C )− 2iB }dz2

where Λ = Adx2 + 2Bdxdy + Cdy2 is a smooth linear combination Λ = fI + ĝΠ of the fundamental forms on S, and z = x + iy is any conformal parameter on R. Most results deal with the case in which R = RΛ is determined on S by some smooth positive definite linear combination Λ = fI + gII on S. It is shown, for example, that S is isothermal with respect to Λ if and only if RΛ supports a holomorphic ΩΛ0 in some neighborhood of any nonumbilic point. By way of contrast, another result states that a holomorphic ΩΛ0 is automatically available in the neighborhood of any nonumbilic point p, unless R coincides at p with some RΛ. The paper closes with a study of surfaces which support an RΛ on which both ΩI0 and ΩII0 are holomorphic.

Mathematical Subject Classification
Primary: 53.75
Received: 8 January 1968
Published: 1 September 1969
Tilla Weinstein