Variational problems and PDEs in affine differential geometry

Tom 69 / 2005

H. Z. Li Banach Center Publications 69 (2005), 9-41 MSC: 53A15, 49Q05, 53C42. DOI: 10.4064/bc69-0-1


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.


  • H. Z. LiDepartment of Mathematical Sciences
    Tsinghua University,
    Beijing, 100084, People's Republic of China

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