Non-Positive Curvature and Global Invertibility of Maps

Li, Gang; Xavier, Frederico

doi:10.1007/s12220-012-9368-3

Cited by 5 publications

(5 citation statements)

References 31 publications

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“…The proof of Xavier and Nollet is based on arguments involving degree theory and cannot be extended in a general form to the infinite-dimensional setting, only for restricted classes of maps. Note that if for all v = 0, Some recent extensions of this kind of theorem for finite-dimensional manifolds can be found in [36]. So, a pertinent question is whether we can replace all of the above metric conditions in terms of µ(x) by a family of metric conditions with parameter v = 0 in terms of |df (x) * v * | in the finite-dimensional case.…”

Section: 23mentioning

confidence: 99%

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On global inverse theorems

Gutú¹

2016

TMNA

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Abstract. Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of f at every x, namely, f is a local homeomorphism or f is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of f , ideally expecting to conclude that f is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of Graves Theorem is obtained for a suitable submersion f with image in a Banach space: given r > 0 and x 0 in the domain of f we give a radius ̺(r) > 0, closely related to the hypothesis of the Hadamard Theorem, such that B̺(f (x 0 )) ⊂ f (Br (x 0 )).

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Section: 23mentioning

confidence: 99%

“…For functions between Euclidean spaces of the same dimension, Nollet and Xavier [43] improved the Hadamard integral condition using the Palais-Smale condition but with dynamical systems techniques. This work was recently extended in [36] for finite-dimensional manifolds.…”

mentioning

confidence: 99%

On global inverse theorems

Gutú¹

2016

TMNA

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show abstract

“…Some recent extensions of this kind of theorem for finite-dimensional manifolds can be found in [36]. So, a pertinent question is whether we can replace all of the above metric conditions in terms of µ(x) by a family of metric conditions with parameter v = 0 in terms of |df (x) * v * | in the finite-dimensional case.…”

Section: 23mentioning

confidence: 99%

mentioning

confidence: 99%

On global inverse and implicit functions

Gutú

2015

Preprint

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Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of f at every x, namely, f is a local homeomorphism or f is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of f , ideally expecting to conclude that f is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of Graves Theorem is obtained for a suitable submersion f with image in a Banach space: given r > 0 and x 0 in the domain of f we give a radius ̺(r) > 0, closely related to the hypothesis of the Hadamard Theorem, such that B̺(f (x 0 )) ⊂ f (Br (x 0 )).

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“…This can be seen in the circle of ideas that originated in [27] and [7], starting from simple covering spaces arguments (as in [40]), and leading to: degree theory and the Palais-Smale condition in [37]; foliations and intersection numbers in [3]; the Hopf fibration and the distortion theorems from the theory of univalent functions in [36]; dynamics and horocycles in geometries of negative curvature in [9], [16], [34] and [47]; conformal geometry, partial differential equations and the Poincaré-Hopf theorem in the present paper. Needless to say, other authors have also made contributions from a topological standpoint to the global invertibility program (see, for instance, [41], [42]).…”

Section: Introductionmentioning

confidence: 99%

Conformal geometry, Euler numbers, and global invertibility in higher dimensions

Xavier

2020

Preprint

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It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar technique one recovers the result that a polynomial local biholomorphism of complex n-space into itself is invertible if and only if the pull-back of every complex line is a connected rational curve. These results are special cases of our main theorem, whose proof uses geometry, complex analysis, elliptic partial differential equations, and topology.

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Non-Positive Curvature and Global Invertibility of Maps

Cited by 5 publications

References 31 publications

On global inverse theorems

On global inverse theorems

On global inverse and implicit functions

Conformal geometry, Euler numbers, and global invertibility in higher dimensions

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