MORSE FUNCTIONS WITH C^{1,1} REGULARITY

Abstract

Milnor’s Lectures on the h-cobordism theorem [7], Theorem 2.7 states: If M is a compact manifold without boundary, the Morse functions form an open dense subset of C∞ (M, R) in the C2 topology. In this thesis, our main work is generalizing this theorem. We build up the weakly Morse function and show that the set of weakly Morse functions is open and dense in C^ {1,1} (T^n , R). In order to prove the Main Theorem, this thesis is divided into five chapters. Chapter 1 provides an introduction to Morse theory and the motivation behind this problem. Chapters 2 and 3 present important properties of Morse functions and Lipschitz functions that are needed throughout the thesis. In Chapter 4, we list and prove several important results related to convex hulls. In particular, Carathéodory’s Theorem 4.5 on convex hulls, Theorem 4.6 on compact preserving property of the convex hull operator, and Theorem 4.10 and 4.12 will provide the necessary tools to separate compact sets away from closed sets. In the final chapter, we use the theorems discussed earlier to establish a series of containment relationships and apply the Clarke Inverse Function Theorem 5.10 to prove the main theorem 5.15.