Index and topology of minimal hypersurfaces in $$\mathbb {R}^n$$ R n

Abstract

In this paper, we consider immersed two-sided minimal hypersurfaces in $$\mathbb {R}^n$$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $$n=4$$ , we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective Morse index bound by purely topological invariants, and is a generalization of Li and Wang (Math Res Lett 9(1):95–104, 2002). Using our index estimates and ideas from the recent work of Chodosh–Ketover–Maximo (Minimal surfaces with bounded index, 2015. arXiv:1509.06724 ), we prove compactness and finiteness results of minimal hypersurfaces in $$\mathbb {R}^4$$ with finite index.