Some geometric problems involving conformal deformation of metrics.

摘要

Given an n-dimensional Riemannian manifold M with metric g and a positive function u defined on M, the scalar curvature of the metric u4n-2 is given by Ru4n-2 g=-cn -1u-n+2n-2 Dgu-cnR gu where cn=n-24 n-1. In Chapter 1, we study the case where M = S3 and the scalar curvature of the metric g is positive. We prove that for any r > ⅔, there exists a constant which only depends on the lower bound on the total volume, the upper bound on the diameter, the lower bound on the Ricci curvature and r, such that if Rg LrS3 is smaller than this constant, then we can deform g by the Green's function G of the conformal Laplacian 8Delta g - R(g) at a point P ∈ S3 such that the asymptotically flat and scalar flat manifold (S3{lcub} P{rcub}, G4g) contains a minimal surface.; In Chapter 2, we study the case where (M, g) is a locally conformally flat compact manifold. When the scalar curvature R( g) = 0 and the dimension of M is 3 or 4, let K:=&cubl0;K: K0somewhereon M,M Kdvg≤-1CK 0,and ∥K∥C3 ≤CK&cubr0; where CK is some constant. For any function K ∈ K , if u is a positive solution of the equation Dgu+Kup=01+z ≤p≤n+2n-2 with bounded energy E(u) Λ, then there are uniform upper and lower bounds on its C3-norm. The bounds only depend on M, g, C K, Λ and z . This a priori estimate generalizes the energy estimates for minimizing solutions derived by Escobar and Schoen to prove an existence theorem for the equation Dgu+Kun+2n-2 =0. Similar techniques can also be applied to show that when M is 4-dimensional with positive scalar curvature and K > 0 on M, then the solutions of Deltagu - R(g)u + Kup = 0 (1 + z p n+2n-2 ) can only have simple blow up points.