Volume 23, issue 7 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 7, 3233–3749
Issue 6, 2701–3231
Issue 5, 2165–2700
Issue 4, 1621–2164
Issue 3, 1085–1619
Issue 2, 541–1084
Issue 1, 1–540

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Subscriptions
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Paul M N Feehan

Geometry & Topology 23 (2019) 3273–3313
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 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.

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