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Elliptic Systems

  • Steffen Fröhlich
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2053)

Abstract

This is an intermediate chapter which first introduces into the theory of non-linear elliptic systems with quadratic growth in the gradient, and which presents secondly some results concerning curvature estimates and theorems of Bernstein-type for surfaces in Euclidean spaces of arbitrary dimensions.A famous result of S. Bernstein states that a smooth minimal graph in \({\mathbb{R}}^{3},\) defined on the whole plane \({\mathbb{R}}^{2},\) must necessarily be a plane. Today we know various strategies to prove this result, and the idea goes back to E. Heinz to establish first a curvature estimate and to deduce Bernstein’s result in a second step. However, minimal surfaces with higher codimensions do not share this Bernstein property, as one of our main examples \(X(w) = (w,{w}^{2}) \in {\mathbb{R}}^{4}\) with \(w = u + iv\) convincingly shows. It is still a great challenge to find geometrical criteria, preferably in terms of the curvature quantities of the surfaces’ normal bundles, which guarantee the validity of Bernstein’s theorem.We must admit that we can only discuss briefly some points where we would wish to employ our tools we develop in this book, but up to now we can not continue to drive further developments.

Keywords

Minimal Surface Unit Normal Vector Normal Bundle Curvature Estimate Curvature Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Steffen Fröhlich
    • 1
  1. 1.Department of Physics, Mathematics, Computer SciencesJohannes Gutenberg UniversityMainzGermany

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