Let (M, g) be a compact Riemannian manifold with boundary. We address the Yamabe-type problem of finding a conformal scalar-flat metric on M whose boundary is a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some of the remaining cases of this problem.

We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresponding Euclidean equations. * Supported by CAPES and FAPERJ (Brazil).

Let (M, g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary. 4 n−2 g with scalar * Supported by CNPq/Brazil grant 309007/2016-0 and CAPES/Brazil grant 88881.169802/2018-01. † Supported by CNPq/Brazil grant 310983/2017-7. arXiv:1807.08406v2 [math.DG] 21 Apr 2019 In the case of manifolds without boundary, the question of compactness of the full set of smooth solutions to the Yamabe equation was first raised by R.Schoen in a topics course at Stanford University in 1988. A necessary condition is that the manifold M n is not conformally equivalent to the sphere S n . This problem was studied in [12,13,21,22,24,25,28,30] and was completely solved in [6,8,20]. In [6], Brendle discovered the first smooth counterexamples for dimensions n ≥ 52 (nonsmooth examples were obtained by Ambrosetti and Malchiodi in [5]). In [20], Khuri, Marques and Schoen proved compactness for dimensions n ≤ 24. Their proof contains both a local and a global aspect. The local aspect involves the vanishing of the Weyl tensor up to order [ n−6 2 ] at any blow-up point and the global aspect involves the positive mass theorem. Finally, in [8], Brendle and Marques extended the counterexamples of [6] to the remaining dimensions 25 ≤ n ≤ 51. In the case of nonempty umbilical boundary, the same compactness and noncompactness results were obtained by Disconzi and Khuri in [11] for the boundary condition B g u = 0.Despite its additional technical difficulties, the question of compactness of the solutions of (1.1) turns out to have great similarity with the one above for the classical Yamabe equation. In [17] Felli and Ould Ahmedou prove compactness for locally conformally flat manifolds with umbilic boundary, a result previously obtained by Schoen [28] for the classical Yamabe equation. In [1] the first author proves the vanishing of the trace-free boundary second fundamental form in dimensions n ≥ 7 at any blow-up point, a result inspired by the vanishing of the Weyl tensor in dimensions n ≥ 6 obtained by Li-Zhang and Marques independently in [21,22,25]. On the other hand, the noncompactness results of Brendle and Marques inspired the first author's paper [2] which provides counterexamples in dimensions n ≥ 25 to compactness in (1.1). So Theorem 1.1 ensures that there is a critical dimension 3 < n 0 ≤ 25 such that compactness for the set of positive smooth solutions of (1.1) holds for n < n 0 and fails for n ≥ n 0 .Although the corresponding result for the classical Yamabe equation in dimension 3 was obtained by Li and Zhu in [24], our approach to Theorem 1.1 makes use of some further techniques of the later works [20,25]. This is because the canonical bubble, coming from the Euclidean metric on B 3 , fails to provide a good approximation for the blowing up solutions of (1.1).