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Non-linear Elliptic Equations in Conformal Geometry
Zurich Lectures in Advanced Mathematics

Sun-Yung Alice Chang (Princeton University, USA)

Non-linear Elliptic Equations in Conformal Geometry

ISBN print 978-3-03719-006-7, ISBN online 978-3-03719-506-2
DOI 10.4171/006
October 2004, 100 pages, softcover, 17.0 x 24.0 cm.
24.00 Euro

Non-linear elliptic partial differential equations are an important tool in the study of Riemannian metrics in differential geometry, in particular for problems concerning the conformal change of metrics in Riemannian geometry. In recent years the role played by the second order semi-linear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully non-linear elliptic equations associated with other symmetric functions of the Ricci tensor. A case of particular interest is the second symmetric function of the Ricci tensor in dimension four closely related to the Pfaffian.

In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g. higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four.

Further Information

Review in Bull. Amer. Math. Soc. 44 (2007), 323–330

Review in Zentralblatt MATH 1064.53018

Review in MR 2104700 (2005h:53053)