Topology of complete minimal submanifolds in $\mathbb{R^{n+m}}$ with finite total curvature

TL;DR

This paper extends finite diffeomorphism classification results to complete immersed minimal submanifolds of any codimension in Euclidean space with finite total curvature and volume growth.

Contribution

It adapts methods from hypersurface cases to arbitrary codimension minimal submanifolds with finite total curvature.

Findings

Proves finite diffeomorphism types for these submanifolds.

Extends previous hypersurface results to higher codimension.

Provides a framework for classifying minimal submanifolds with finite total curvature.

Abstract

In [CKM17], Chodosh, Ketover, and Maximo proved finite diffeomorphism theorems for complete embedded minimal hypersurfaces of dimension $\leqslant$ 6 with finite index and bounded volume growth ratio. In this paper, we adapt their method to study finite diffeomorphism types for complete immersed minimal submanifolds of arbitrary codimension in Euclidean space with finite total curvature and Euclidean volume growth.