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Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Paul M N Feehan

Geometry & Topology 23 (2019) 3273–3313

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 C1 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 CN and generalized Morse–Bott of order N 3; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is C2 and Morse–Bott on a Banach space.

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