Minimal hypersurfaces of finite $\delta$-index in ${\mathbb{R}}^4$ and ${\mathbb{R}}^5$

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 $\delta$-index, finite Morse index, and finite total curvature are equivalent for $\delta >((n-1)\mathrm{/}n{)}^2$, without assumptions on properness, volume growth, or topology. The main estimate shows that $\delta$-stability outside a compact set and finite-dimensional $H_c^1(M;\mathbb{R})$ imply intrinsic Euclidean volume growth for $\delta >(n-2)\mathrm{/}n$. We also deduce the sharp $\delta$-stable Bernstein theorem in this latter range from results of Hong–Li–Wang and Florit-Simon.