Prescribing the curvature of Riemannian manifolds with boundary

Abstract

Let M be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function f on ∂M (resp. on M ) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on M (resp. metric on M with geodesic boundary). In order to provide analogous results for this problem with n ≥ 3, we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on ∂M (resp. on M ) is a mean curvature of a scalar flat metric on M (resp. scalar curvature of a metric on M and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.

Critical metrics and curvature of metrics with unit volume or unit area of the boundary

“…In others words we can locally prescribe the curvature of such manifolds with constant volume or constant area of the boundary. This will follow from a slight modification of the proof of Proposition 3.1 and 3.3 of [16], see also Section 5 for a more general approach.…”

“…In order to prove (b), we modify slightly the arguments in Proposition 3.1 of [16]. Since V is not identically zero on Σ, it follows from (2.2) that Π g = Hg n−1 g. Suppose that {e i } n−1 i=1 span T Σ locally and consider e n = ν.…”

“…In order to prove our prescribing curvature results, first we observe that part of techniques used in beginning of this subsection has been introduced in [16]. Thus, in order to help the reader and for the sake of clarity, we recall, when needed, the appropriately modified statements of the theorems we use.…”

“…In [16], in order to prescribe the curvature of manifolds with boundary, the first named author and F. Vitório proved that the map g → (R g , 2H g ) is almost always a surjection, where R g and H g denote the scalar curvature and the mean curvature of the boundary of the metric g, respectively. In this work we study this same map under the effect of the volume and the area of the boundary through two approaches: on the one hand, we vary the volume or area of the boundary and, on the other hand, we restrict it to metrics of unit volume or unit area of the boundary.…”

“…We make use of the Yamabe invariant and of the variational characterization of the map g → (R g , 2H g ) to prescribe the curvature of manifolds with boundary under some constraints on the volume or boundary area (for cases without any constraint see for instance [16,20,30,31]). This problem was first approached by O. Kobayashi [27] in the closed case, which was based in the Kazdan-Warner problem [25,26] of finding metrics with prescribed scalar curvature.…”