A positive energy theorem for fourth-order gravity

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

“…The above theorem represents the exact analogue of the results in [22] in the AE setting and in the context of Q-curvature invariants. This provides a direct link between the J-tensor at infinity and the fourth order energy (10), which from [3] we know to be related to several rigidity phenomena associated to Q-curvature. For instance, E(g) is positively proportional to the mass of the Paneitz operator studied in [24,20,21].…”

“…The above multiplication property can be deduced using the tools developed in [5], and can also be found in [7, Lemma 5.5] and the corresponding L 2 -version can also be found in [13,Lemma 2.5]. 3 Below, we will now establish a multiplication property which is tailored for some of our specific applications.…”

“…7 In parallel to the above comments, E(g) was introduced in [4] as a quantity defined on AE-slices of asymptotically Minkowskian space-time solutions to a set of fourth order equations, which is conserved through time translations. In particular, from the work of [4] and [3], the Q-curvature of the corresponding fourth order initial data set appears as a natural analogue to the scalar curvature for the corresponding relativistic initial data sets. Thus, E g (1) appears as the corresponding dynamical equivalent of the global Lagrangian approach defining E(g) in [4], with the same conserved energy on the space slices.…”

“…where J g is defined by (7) and the Q-curvature by (3). From [26,2], it is known to satisfy the local conservation identity div g G J = 0.…”

“…In [3][Theorem A], the above notion of energy was analysed in detail and the following positive energy theorem was established Theorem 1.1 (Positive Energy [3]). Let (M n , g) be an n-dimensional AE manifold, with n ≥ 3, which satisfies the decaying conditions: (i)…”