Global inversion via the Palais-Smale condition

Scott Nollet; Frederico Xavier

doi:10.3934/dcds.2002.8.17

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2002, Volume 8, Issue 1: 17-28. Doi: 10.3934/dcds.2002.8.17

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Global inversion via the Palais-Smale condition

Scott Nollet^1, and
Frederico Xavier^2,

1.
Department of Mathematics, Texas Christian University, Fort Worth, TX 76129
2.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

Received: February 28, 2001

Revised: September 30, 2001

Published: December 2001

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Abstract

Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.

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Mathematics Subject Classification: 58F25, 58C15, 14R15.

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