On the Structure of Compact Static Vacuum Spaces and Positive Isotropic Curvature

초록

In this paper, we study geometric structures of compact static vacuum spaces and function theoretic properties for the potential functions satisfying the static vacuum equation. In particular, we investigate geometric conditions under which static vacuum spaces are warped product and Bach-flat. As an application, we prove that if a triple (Mn, g, f ), n ≥ 4, is a compact static vacuum space satisfyingω := d f ∧i∇ f Ric˚ = 0, then M is either isometric to a round sphere or a warped product of a circle with a compact Einstein manifold of positive Ricci curvature, up to finite cover. Furthermore, if (M, g) has positive isotropic curvature, then M is either isometric to a round sphere or a product S1 × Sn−1.

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