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

Qi Ding, Lei Zhang

TL;DR

This work extends finiteness results for minimal hypersurfaces to complete immersed minimal submanifolds of arbitrary codimension in Euclidean spaces with finite total curvature and Euclidean volume growth, establishing a uniform bound on diffeomorphism types. The authors develop extrinsic and intrinsic curvature estimates under small total curvature, introduce smooth blow-up sets to analyze concentration, and study the ends of such submanifolds, proving ends are regular at infinity with precise asymptotics governed by the linearized operator. A central contribution is proving finiteness of topology via an annular decomposition and an induction on total curvature, generalizing Anderson-type compactification arguments to immersed higher-codimension settings. The results provide a robust quantitative framework for understanding the topology of finite-total-curvature minimal submanifolds and offer techniques applicable to related geometric analysis problems.

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.

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

TL;DR

This work extends finiteness results for minimal hypersurfaces to complete immersed minimal submanifolds of arbitrary codimension in Euclidean spaces with finite total curvature and Euclidean volume growth, establishing a uniform bound on diffeomorphism types. The authors develop extrinsic and intrinsic curvature estimates under small total curvature, introduce smooth blow-up sets to analyze concentration, and study the ends of such submanifolds, proving ends are regular at infinity with precise asymptotics governed by the linearized operator. A central contribution is proving finiteness of topology via an annular decomposition and an induction on total curvature, generalizing Anderson-type compactification arguments to immersed higher-codimension settings. The results provide a robust quantitative framework for understanding the topology of finite-total-curvature minimal submanifolds and offer techniques applicable to related geometric analysis problems.

Abstract

In [CKM17], Chodosh, Ketover, and Maximo proved finite diffeomorphism theorems for complete embedded minimal hypersurfaces of dimension 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.
Paper Structure (9 sections, 12 theorems, 99 equations)

This paper contains 9 sections, 12 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notation and Preliminaries
  4. Definitions and basic notation
  5. Curvature estimates.
  6. Geometry of Minimal Submanifolds with Finite Total Curvature
  7. Finiteness of Topology
  8. Calculations of Minimal Surface System
  9. Topology Lemma on Covering Space

Key Result

Theorem 1.1

For fixed $n\,,m \in \mathbb{Z}^+\,,n \geqslant 3\,, m \geqslant1\,,$ and $\Gamma \, ,\Lambda \in \mathbb{R} \,, \Gamma\,,\Lambda \geqslant 0\,$, there exists $N = N(n,m,\Gamma,\Lambda) \in \mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1: Curvature estimates in the extrinsic distance
  • proof
  • Remark 2.2
  • Lemma 2.3: Curvature estimates in the intrinsic distance
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 15 more