Some geometric properties of generalized $$\varphi $$-vacuum static spaces

Abstract In this paper we study the geometry of generalized $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -vacuum static spaces, proving estimates for the $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -scalar curvature and for the first eigenvalue of the Jacobi operator, and also rigidity under various geometric ass be a compact manifold. umptions; in particular, we prove a result related to the famous Cosmic no-hair conjecture of Boucher, Gibbons and Horowitz.