On surfaces with prescribed shape operator
Abstract
The problem of immersing a simply connected surface with a prescribed shape operator
is discussed. I show 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.
Type
Journal articlePermalink
https://hdl.handle.net/10161/12687Collections
More Info
Show full item recordScholars@Duke
Robert Bryant
Phillip Griffiths Professor of Mathematics
My research concerns problems in the geometric theory of partial differential equations.
More specifically, I work on conservation laws for PDE, Finsler geometry, projective
geometry, and Riemannian geometry, including calibrations and the theory of holonomy.
Much of my work involves or develops techniques for studying systems of partial differential
equations that arise in geometric problems. Because of their built-in invariance
properties, these systems often have specia

Articles written by Duke faculty are made available through the campus open access policy. For more information see: Duke Open Access Policy
Rights for Collection: Scholarly Articles
Works are deposited here by their authors, and represent their research and opinions, not that of Duke University. Some materials and descriptions may include offensive content. More info