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On the Bumpy Metrics Theorem for Minimal Submanifolds

Brian White

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.

On the Bumpy Metrics Theorem for Minimal Submanifolds

Abstract

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

Paper Structure

This paper contains 4 sections, 8 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Generalizations
  4. An open problem

Key Result

Theorem 2.1

Let $N$ be a smooth manifold. Let $G$ be a finite group of diffeomorphisms of $N$. Suppose that $k$ is an integer $\ge 3$ or that $k=\infty$. Then a generic, $G$-invariant, $C^k$ Riemannian metric on $N$ is bumpy in the following sense: no closed, minimal immersed submanifold $M$ of $N$ has a nontri

Theorems & Definitions (19)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3: Structure Theorem
  • proof
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 9 more