CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Some Rigidity Results Related to Monge—Ampère Functions

  Published:2009-12-04
 Printed: Apr 2010
  • Robert L. Jerrard
Format:   HTML   LaTeX   MathJax  

Abstract

The space of Monge-Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2 u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampère functions. We also prove that if a Monge-Ampère function $u$ on a bounded set $\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge-Ampère function of $2$ variables.
MSC Classifications: 49Q15, 53C24 show english descriptions Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]
Rigidity results
49Q15 - Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]
53C24 - Rigidity results
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/