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arXiv · 2609.14317

Minimal hypersurfaces of finite $δ$-index in $\mathbb R^4$ and $\mathbb R^5$

Marcos Ranieri
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Abstract

Let $X:M^n\to\mathbb R^{n+1}$ be a complete, connected, two-sided minimal immersion without boundary, where $n=3,4$. We prove that finite $δ$-index, finite Morse index, and finite total curvature are equivalent for $δ>((n-1)/n)^2$, without assumptions on properness, volume growth, or topology. The main estimate shows that $δ$-stability outside a compact set and finite-dimensional $H_c^1(M;\mathbb R)$ imply intrinsic Euclidean volume growth for $δ>(n-2)/n$. We also deduce the sharp $δ$-stable Bernstein theorem in this latter range from results of Hong--Li--Wang and Florit-Simon.

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Marcos Ranieri. 2026-09-13. Minimal hypersurfaces of finite $δ$-index in $\mathbb R^4$ and $\mathbb R^5$. https://arxiv.org/abs/2609.14317

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