Compacitification and Positive Mass Theorem for Fibered Euclidean End

Abstract

We consider the Positive Mass Theorem for Riemannian manifolds $$(M^{n},g)$$ ( M n , g ) with the asymptotic end $$(\mathbb {R}^{k}\times X^{n-k}, g_{\mathbb {R}^{k}}+g_{X})$$ ( R k × X n - k , g R k + g X ) ($$k\ge 3$$ k ≥ 3 ) by studying the corresponding compactification problem. Here $$(X, g_X)$$ ( X , g X ) is a compact scalar flat manifold. We show that the Positive Mass Theorem holds if certain generalized connected sum admits no metric of positive scalar curvature. Moreover we establish the rigidity result, namely, the mass is zero iff M is isometric to $$\mathbb {R}^{k}\times X^{n-k}$$ R k × X n - k .