Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Ranieri, Marcos
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213292220