ConstantT-curvature conformal metrics on 4-manifolds with boundary

Abstract

In this paper we prove that, given a compact four-dimensional smooth Riemannian manifold (M, g) with smooth boundary, there exists a metric conformal to g with constant T -curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth-order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a third-order pseudodifferential operator and homogeneous Neumann operator. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from below, we look for saddle points. We do this by using topological arguments and min-max methods combined with a compactness result for the corresponding BVP.

Integral pinching results for manifolds with boundary

“…The second inequality that we are going to state is a trace analogue of the previous one. Its proof can be found [32]. Proposition 2.8.…”

Prescribing the Q-curvature on the sphere with conical singularities

“…The complication of the problem increases considerably in odd dimensions since the operator becomes a pseudo differential one (the principal part is of the form (−∆ g ) n 2 ). Even though the regular problem was studied extensively, see [31], [35] and the references therein, to the best of our knowledge, the singular one was not treated for dimensions greater than 2. Also we cannot use the approach developed in [12] and [13] in dimension 2, since we do not have a similar singular Moser-Trudinger inequality (a proof of such inequality would lead to a better understanding of the problem).…”

Q-curvature flow on 4-manifolds with boundary

“…Suppose that the assumptions of Theorem 1.1 hold, and let u(t) be the unique local solution to the initial scalar parabolic boundary value problem (14). Then for every…”

“…Suppose that the assumptions of Theorem 1.1 holds and let u(t) be the unique local solution to the initial boundary value problem corresponding to (14). Then for every T > 0 such that u is defined on [0, T [, and and for every positive (relative) integer k ≥ 3 there exists C = C(T, k) > 0 such that…”