A positive mass theorem for asymptotically flat manifolds with a non-compact boundary

Abstract

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.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

“…have played an important role in such prescribed conformal curvature problems. As a direct consequence of Theorem 1.3 and the positive mass type theorem recently proved in [3], we obtain Theorem 1.4. Let (M, g 0 ) be a smooth compact Riemannian manifold of dimension n ≥ 3 with boundary.…”

“…The definition of the mass m(g) was first proposed by Marques. The following positive mass type conjecture was initially given in [2] and has been verified in [3,Theorem 1.3] under the hypotheses that either 3 ≤ n ≤ 7 or if n ≥ 3 and N is spin.…”

Capacity inequalities and rigidity of cornered/conical manifolds

“…The above definition is independent of the particular choice of the chart at infinity what means that the mass is a geometric invariant. As already mentioned, we also have a Positive mass theorem in this context (see [1], Theorem 1.1), which states that an asymptotically flat manifold with non-negative scalar curvature and mean convex boundary has nonnegative mass if either 3 ≤ n ≤ 7 or n ≥ 3 and M is spin. Moreover the mass is zero if only if it is isometric to R n + with the flat metric.…”

“…The hypersurface flows in the outward normal direction with speed N = 1 H and it is easy to see that there is a problem if either H = 0 or H changes the sign on Σ t . 1 To overcome theses problems, Marquadt [20,22] developed the notion of weak solutions for (2.2), proving the existence and uniqueness of such solutions guided by the ideas of Huisken and Ilmanen [12].…”

The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds

“…exists, we call it the mass of (N, g). The following positive mass type conjecture has been verified by Almaraz-Barbosa-de Lima [3], provided that 3 ≤ n ≤ 7 or n ≥ 8 and N is spin.…”