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The local Morse Homology of the critical points in the Lagrange problem

Xiuting Tang

Abstract

In this paper, we construct local Morse homology in a new way and compute the local Morse homology of the critical points of the Lagrange problem. As a corollary, we prove for the first time that each of the linear critical points is either a saddle point or a degenerate critical point compared to the previous conclusion that if all the linear critical points are non-degenerate, they are saddle points.

The local Morse Homology of the critical points in the Lagrange problem

Abstract

In this paper, we construct local Morse homology in a new way and compute the local Morse homology of the critical points of the Lagrange problem. As a corollary, we prove for the first time that each of the linear critical points is either a saddle point or a degenerate critical point compared to the previous conclusion that if all the linear critical points are non-degenerate, they are saddle points.
Paper Structure (10 sections, 32 theorems, 222 equations)

This paper contains 10 sections, 32 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Local Morse homology
  3. Introduction of local Morse homology
  4. Local behavior of the gradient flow lines near critical points
  5. Compactness of the Moduli space of the gradient flow lines
  6. Transversality and the moduli space of the gradient flow lines as a manifold
  7. Definition of local Morse homology
  8. Invariance of local Morse homology
  9. Applications in the Lagrange problem
  10. Acknowledgments

Key Result

Lemma 1

For any $\eta>0$, there exists $\nu_0$, such that when $\nu>\nu_0$, we have $\max f_\nu|_{C_\nu}-\min f_\nu|_{C_\nu}<\eta$, where $C_\nu=\{x\in B_\delta|\nabla f_\nu (x)=0\}$.

Theorems & Definitions (60)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 3
  • ...and 50 more