Variational problems and PDEs in affine differential geometry

H. Z. Li

Banach Center Publications (2005)

  • Volume: 69, Issue: 1, page 9-41
  • ISSN: 0137-6934

Abstract

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This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke [Bla], Chern [Ch], C. P. Wang [W], Li-Li-Simon [LLS], and Calabi [Ca-II]. We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate Bernstein problems and give partial solutions.

How to cite

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H. Z. Li. "Variational problems and PDEs in affine differential geometry." Banach Center Publications 69.1 (2005): 9-41. <http://eudml.org/doc/282518>.

@article{H2005,
abstract = {This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke [Bla], Chern [Ch], C. P. Wang [W], Li-Li-Simon [LLS], and Calabi [Ca-II]. We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate Bernstein problems and give partial solutions.},
author = {H. Z. Li},
journal = {Banach Center Publications},
keywords = {equiaffine maximal hypersurfaces; centroaffine extremal hypersurfaces; affine extremal graph; Bernstein problems; affine spheres},
language = {eng},
number = {1},
pages = {9-41},
title = {Variational problems and PDEs in affine differential geometry},
url = {http://eudml.org/doc/282518},
volume = {69},
year = {2005},
}

TY - JOUR
AU - H. Z. Li
TI - Variational problems and PDEs in affine differential geometry
JO - Banach Center Publications
PY - 2005
VL - 69
IS - 1
SP - 9
EP - 41
AB - This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke [Bla], Chern [Ch], C. P. Wang [W], Li-Li-Simon [LLS], and Calabi [Ca-II]. We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate Bernstein problems and give partial solutions.
LA - eng
KW - equiaffine maximal hypersurfaces; centroaffine extremal hypersurfaces; affine extremal graph; Bernstein problems; affine spheres
UR - http://eudml.org/doc/282518
ER -

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