Math @ Duke

Publications [#243381] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, On surfaces with prescribed shape operator,
Results in Mathematics, vol. 40 no. 14
(2001),
pp. 88121, Springer Nature (Dedicated to ShiingShen Chern on his 90th birthday.) [MR2002i:53010], [math.DG/0107083], [doi]
(last updated on 2019/05/22)
Author's Comments: Please note: The published version is, by mistake,
a preliminary version. The correct version is the one
posted on the arXiv.
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.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

