Minimization of curvature in Conformal Geometry

Abstract

Let (M, g0) a smooth compact Riemannian manifold with smooth boundary and dimension n ≥ 3. We consider a minimization problem for the scalar curvature R after a conformal change of the form g = u2∗−2g0, where u is a smooth positive function on M . In particular, we seek for minimizers of the || · ||∞ functional of R, within a conformal class, under small energy assumptions and natural geometric constraints, in order to generalize a result of Moser and Schwetlick in the case of surfaces with boundary.

The nonreflexiveness of the space L∞(M ) forces us to first study the corresponding minimization problem for the Lp norm of R after a conformal change. We establish the existence of a priori bounds for solutions of our equation under our assumptions. Then we prove the existence of minimizers for the assosciated p-problem via the Direct Method. The 4th-order Euler Lagrange equations are derived and studied, as well as regularity properties of their solutions, for all p ∈ (q, ∞), where q > n/2 .

Finally we let p → +∞ and show that the limit equation produces a minimizer for our original problem. Moreover we study the structure of the nodal set Γ of a solution of the limit Euler Lagrange equation. We draw a connection between the form of the curvature of the minimizer and this nodal set. We specifically show, that the minimizer will have constant scalar curvature, outside of the set Γ, thus obtaining a connection with the Yamabe Problem on manifolds with boundary.

Date of Award	18 Dec 2015
Original language	English
Awarding Institution	University of Bath
Supervisor	Roger Moser (Supervisor)