The scalar curvature problem on the four dimensional half sphere

Abstract

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature for some metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

“…critical point z of K 1 with (∂K/∂ν)(z) > 0 and critical point y of K. In this paper, we generalize the results of [10]. We cancel the assumption of [10] which is (∂K/∂ν) < 0 at each critical point of K 1 that is the function…”

“…The goal of our first result is the construction of some solutions (u ε ) of (P ε ) which blow up at two different points, one of them lies on the boundary and the other is an interior point. This result implies that there are more critical points at infinity than the authors found in [10]. Before stating the result, we need to introduce some notations.…”

“…Their approach involves a fine blow up analysis of some subcritical approximations and the use of the topological degree tools. Ben Ayed-El Mehdi-Ould Ahmedou [9], [10] gave some topological conditions on K to prescribe the scalar curvature under minimal boundary conditions on half spheres of dimension n ≥ 4. Note that problem (P ) is in some sense related to the well known scalar curvature problem −Δ g u + (n(n − 2)/4)u = Ku (n+2)/(n−2) , u > 0 in S n (3) to which much work has been devoted (see [3,4,7,13,14,[19][20][21] and the references therein).…”

“…Hence, the tower bubble solutions do not exist. Moreover Ben Ayed et al [10] proved that there are critical points at infinity (following the terminology of A. Bahri) for the functional J 0 (defined by (1.2)) associated to the problem (P ). Recall that the critical point at infinity is the limit of the orbits of the gradient flow of J 0 which remain in V (p, η(s)), where η(s) is a given function which tends to 0 as s tends to infinity and V (p, η) is defined by (3.1) below (for more details, see Definition 4.1 below).…”

“…, y p ) where the points y i 's are different critical points of K in S 4 + . Note that, in [10], the authors assumed that (∂K/∂ν) < 0 at each critical point of K 1 . This assumption allows them to cancel the influence of the boundary.…”