Non-linear Elliptic Equations in Conformal GeometryNon-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. The material is suitable for graduate students and research mathematicians interested in geometry, topology, and differential equations. |
Contents
Preface | 5 |
Polyakov formula on compact surfaces | 17 |
Extremal metrics for the logdeterminant functional | 38 |
Elementary symmetric functions | 50 |
A priori estimates for the regularized equation s | 56 |
Smoothing via the Yamabe flow | 74 |
Common terms and phrases
4-manifold Aow)² dvo assume assumption background metric go bounded Branson C(go C₁ C₂ Chapter compact surface conformal change conformal geometry conformal Laplacian conformal transformation conformally covariant operators conformally equivalent conformally invariant const constant Corollary 1.7 denotes duge dvgc dvge dvgw E(uo e²w e2w go eigenvalues elliptic equality iff equivalent to S4 Euler-Lagrange equation FA[w formula Gaussian curvature Gaussian curvature equation Harnack inequality hence Hölder's inequality II[w implies inf E(u integral Lemma manifold metric g Notice Obata's obtain Onofri Paneitz operator proof of Theorem Proposition 1.4 Qo dvo R³ dv recall Rellich's Theorem Remark Ricci tensor satisfies scalar curvature solution tensor terms of g Theorem 5.4 vol(M VR²