On the Bumpy Metrics Theorem for Minimal Submanifolds

TL;DR

This paper extends the Bumpy Metrics Theorem to broader contexts, showing that for generic smooth metrics, minimal submanifolds with nontrivial Jacobi fields are absent, advancing understanding of minimal submanifold stability.

Contribution

The paper generalizes the Bumpy Metrics Theorem to new settings, providing broader conditions under which minimal submanifolds lack nontrivial Jacobi fields.

Findings

Proves generalized versions of the Bumpy Metrics Theorem.

Shows generic metrics have no minimal submanifolds with nontrivial Jacobi fields.

Enhances understanding of stability of minimal submanifolds.

Abstract

This paper proves several natural generalizations of the theorem that for a generic, $C^k$ Riemannian metric on a smooth manifold, there are no closed, embedded, minimal submanifolds with nontrivial jacobi fields.