TY - JOUR

T1 - On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

AU - Feehan, Paul M.N.

N1 - Funding Information:
The author was partially supported by National Science Foundation Grant DMS-1510064 and the Dublin Institute for Advanced Studies.
Funding Information:
I am indebted to Michael Greenblatt and András Némethi for independently pointing out to me that, for functions on Euclidean space, the Morse Lemma for functions with degenerate critical points (also known as the Morse Lemma with parameters or Splitting Lemma) should be the key ingredient needed to prove the main result of this article in the finite-dimensional case (Corollary ). I am extremely grateful to Brian White for explaining results of his [–] and others on minimal surfaces and integrability of Jacobi fields and to Leon Simon for explaining his results and results with Adams on integrability of Jacobi fields in [, , ]. I also thank Carles Bivià-Ausina, Otis Chodosh, Tristan Collins, Santiago Encinas, Luis Fernandez, Antonella Grassi, David Hurtubise, Johan de Jong, Daniel Ketover, Qingyue Liu, Doug Moore, Yanir Rubinstein, Siddhartha Sahi, Ovidiu Savin, Peter Topping, Graeme Wilkin, and Jarek Włodarczyk for helpful communications, discussions, or questions during the preparation of this article. I am grateful to the National Science Foundation for their support and the Dublin Institute for Advanced Studies and Yi-Jen Lee and the Institute of Mathematical Sciences at the Chinese University of Hong Kong for their hospitality and support. Lastly, I am most grateful to the anonymous referee for numerous comments and suggestions that helped improve this article.
Funding Information:
I am indebted to Michael Greenblatt and Andr?s N?methi for independently pointing out to me that, for functions on Euclidean space, the Morse Lemma for functions with degenerate critical points (also known as the Morse Lemma with parameters or Splitting Lemma) should be the key ingredient needed to prove the main result of this article in the finite-dimensional case (Corollary?3). I am extremely grateful to Brian White for explaining results of his [125 ?127] and others on minimal surfaces and integrability of Jacobi fields and to Leon Simon for explaining his results and results with Adams on integrability of Jacobi fields in [4 , 108 , 109]. I also thank Carles Bivi?-Ausina, Otis Chodosh, Tristan Collins, Santiago Encinas, Luis Fernandez, Antonella Grassi, David Hurtubise, Johan de Jong, Daniel Ketover, Qingyue Liu, Doug Moore, Yanir Rubinstein, Siddhartha Sahi, Ovidiu Savin, Peter Topping, Graeme Wilkin, and Jarek W?odarczyk for helpful communications, discussions, or questions during the preparation of this article. I am grateful to the National Science Foundation for their support and the Dublin Institute for Advanced Studies and Yi-Jen Lee and the Institute of Mathematical Sciences at the Chinese University of Hong Kong for their hospitality and support. Lastly, I am most grateful to the anonymous referee for numerous comments and suggestions that helped improve this article.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a C2 Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse–Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Łojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space (Łojasiewicz in Ensembles Semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 1965) and the Morse Lemma (Theorems 4 and 5) for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Łojasiewicz–Simon gradient inequalities for analytic functions on Banach spaces (Theorems 9 and 10).

AB - It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a C2 Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse–Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Łojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space (Łojasiewicz in Ensembles Semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 1965) and the Morse Lemma (Theorems 4 and 5) for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Łojasiewicz–Simon gradient inequalities for analytic functions on Banach spaces (Theorems 9 and 10).

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U2 - 10.1007/s00526-020-01734-4

DO - 10.1007/s00526-020-01734-4

M3 - Article

AN - SCOPUS:85083632035

SN - 0944-2669

VL - 59

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

M1 - 87

ER -