Index and topology of minimal hypersurfaces in 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 generalization of Li-Wang. Using our index estimates and ideas from the recent work of Chodosh-Ketover-Maximo, we prove compactness and finiteness results of minimal hypersurfaces in ${\mathbb{R}}^4$ with finite index.

Various typos corrected, appendix added. Accepted by Calculus of Variation and PDE

References in corpus (3)

• Minimal hypersurfaces with bounded index
• On the topology and index of minimal surfaces
• Higher dimensional Schwarz's surfaces and Scherk's surfaces

Cited by in corpus (2)

• Minimal hypersurfaces with bounded index
• A note on the index of closed minimal hypersurfaces of flat tori