Carnegie Mellon
Department of Mathematical Sciences

Mohammad Reza Pakzad, Department of Mathematics, University of British Columbia.

"Rigidity and Regularity properties of Sobolev isometric immersions"

Abstract

There are several motivations to study the isometric immersions with Sobolev type regularity of say an $ m$-dimensional domain into a given Euclidean space. One motivation is geometrical. It is well known that $ C^2$ isometric immersions have a good classification and enjoy strong rigidity properties while the celebrated results of Nash and Kuiper show that $ C^1$ isometric immersions can be much more complicated (e.g. the image of $ S^2$ can be contained in an arbitrarily small ball). One may consider now the Sobolev classes of maps which lie somewhat in between. On the other hand, spaces of this type arise in the elasticity theory of plates, first formulated by Kirchhoff, and give rise to new questions which can be formulated for higher dimensional sheets. The main example disussed in this talk is the space of isometric immersions $ u$ from a two dimensional domain $ \Omega$ to $ {\mathbb{R}}^3$ which are in the Sobolev class $ W^{2,2}$, i.e. $ \nabla^2 u$ is in $ L^2$.

THURSDAY, December 2, 2004
Time: 1:30 P.M.
Location: PPB 300

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