Scalar curvature rigidity of almost Hermitian manifolds which are asymptotic to CHmCHm

We show that an almost Hermitian manifold (M,g)(M,g) of real dimension 2m   which is strongly asymptotically complex hyperbolic and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space CHmCHm. Assuming Kähler instead of almost Hermitian this gives the already known rigidity results proved for the odd complex dimensional case by M. Herzlich in [M. Herzlich, Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces, Math. Ann. 312 (4) (1998) 641–657] as well as for the even complex dimensional case by H. Boualem and M. Herzlich in [H. Boualem, M. Herzlich, Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces, Ann. Scuola Norm. Sup Pisa (Ser. V) 1 (2) (2002) 461–469].