Some Results on a Fully Nonlinear Equation in Conformal Geometry

Abstract

In this thesis we study the question of solving an equation arising in conformal geometry. Specifically, we obtain information about the equation &sigma;k<\sub>(g<super>&#8722;1<\super>Ag<\sub>) = constant. Here g is a metric in a given conformal class [g0<\sub>] and Ag is the Schouten tensor of g.

First, we generalize a result of Chang, Gursky, and Yang to show that under certain geometric conditions, entire solutions of this equation on R<super>n</super> must necessarily arise via pullback from the sphere under stereographic projection. This &ldquo;Obata&rdquo; type theorem provides an alternate proof of a result of Li and Li. However, our approach uses integral estimates which may be more amenable to application on manifolds which are not locally conformally flat.

Second, we study the question of solving &sigma;k<\sub> = constant on n-manifolds M, the &ldquo;k-Yamabe problem&ldquo;, when k = 2 and n = 3. The k-Yamabe problem has been studied for many values of n and k before (by Li and Li, by Guan and Wang, by Gursky and Viaclovsky, by Sheng, Trudinger, and Wang, and by Trudinger and Wang, among others); however, some cases remain open. The case where n < 2k is unresolved in general. A perhaps more natural question to consider in this case is solving v2k<\sub> = constant, for n < 2k, where v2k<\sub> are the renormalized volume coefficients studied by Graham and by Chang and Fang. Unfortunately, the approach used for other values of n and k does not apply in this final case-for v2k<\sub>, one cannot use the local estimates of Guan and Wang and of Chen. This is because the local estimates use algebraic properties of k which do not hold for v2k<\sub>. Though the &sigma;2<\sub>-Yamabe problem in dimension 3 has been studied before, we study an alternative approach which avoids the use of the local estimates. We hope this approach may prove useful for resolving the question of solving v2k<\sub> = constant.