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On Surfaces with Prescribed Shape Operator

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Abstract

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. It is shown that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

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Correspondence to Robert I. Bryant.

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This article is dedicated to Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research.

The research for this article was begun during a June 2001 conference at the Mathematisches Forschungsinstitut at Oberwolfach. The author thanks the MFO for its hospitality. Thanks also to Duke University for its support via a research grant and to the National Science Foundation for its support via DMS-9870164.

This is Version 2.1. The most recent version can be found at arXiv:math.DG/0107083.

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Bryant, R.I. On Surfaces with Prescribed Shape Operator. Results. Math. 40, 88–121 (2001). https://doi.org/10.1007/BF03322701

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