On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half

TL;DR

This paper establishes that for analytic functions on Banach spaces, having a Lojasiewicz exponent of one half at a critical point implies the function is Morse-Bott, linking gradient inequalities to geometric structure.

Contribution

It proves the converse of the Morse-Bott property for analytic functions on Banach spaces, showing the Lojasiewicz exponent one half characterizes Morse-Bott functions.

Findings

Analytic functions with Lojasiewicz exponent one half are Morse-Bott.

Critical sets are analytic submanifolds when the exponent is one half.

Streamlined proofs of Lojasiewicz-Simon inequalities are provided.

Abstract

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.