Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Feehan, Paul

doi:10.2140/gt.2019.23.3273

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Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Paul M N Feehan

Geometry & Topology 23 (2019) 3273–3313

DOI: 10.2140/gt.2019.23.3273

Abstract

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). Here we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is $C^{1}$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for (real or complex) analytic varieties, that the gradient inequality for an arbitrary analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Łojasiewicz inequalities when a function is $C^{N}$ and generalized Morse–Bott of order $N \geq 3$ ; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is $C^{2}$ and Morse–Bott on a Banach space.

Keywords

analytic varieties, Łojasiewicz inequalities, gradient flow, Morse–Bott functions, resolution of singularities, semianalytic sets and subanalytic sets

Mathematical Subject Classification 2010

Primary: 32B20, 32C05, 32C18, 32C25, 58E05

Secondary: 14E15, 32S45, 57R45, 58A07, 58A35

References

Publication

Received: 31 August 2017

Revised: 8 May 2019

Accepted: 15 June 2019

Published: 30 December 2019

Proposed: Tobias H Colding

Seconded: Bruce Kleiner, Simon Donaldson

Authors

Paul M N Feehan
Department of Mathematics Rutgers, The State University of New Jersey Piscataway, NJ United States

Volume 23, issue 7 (2019)