On Morse Index Retrieval

TL;DR

This paper investigates the Morse index properties of smooth functions near the sphere, proving the existence of functions that admit two indices differing by two, expanding understanding of Morse index extensions.

Contribution

It introduces a novel result showing functions can admit two Morse indices differing by two, which was previously unknown.

Findings

Functions can admit two Morse indices differing by two.

A function cannot admit two indices of different parity.

Theoretical proof of the existence of such functions.

Abstract

A smooth function f in a neighbourhood of the unit sphere $S^{n - 1}$ is said to admit index $\lambda$ if it can be extended to a function F in the unit ball $B^n$ such that F has a unique critical point p and the Morse index of p is equal to $\lambda$. It is easy to see that a function f cannot admit two indices of different parity. We prove that for any two indices that differ by two there exists a function f that admits both of them.