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$L^{2}$ -estimates for the $d$ -equation and Witten’s proof of the Morse inequalities

Bo Berndtsson (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^{2}$ -estimates for the $\bar{\partial}$ -equation is used as motivation. We also use the method to prove $L^{2}$ -estimates for the $d$ -equation with a weight $e^{- t φ}$ where $φ$ is a nondegenerate Morse function.

Lefschetz theorems for the integral leaves of a holomorphic one-form

Carlos Simpson (1993)

Compositio Mathematica

Lemme de Morse et calcul des variations

P. Antoine (1979)

Mémoires de la Société Mathématique de France

Light rays in static spacetimes with critical asymptotic behavior: a variational approach.

Luisi, Valeria (2006)

Electronic Journal of Differential Equations (EJDE) [electronic only]

L'indice de Morse dans le calcul variationnel

J. J. Duistermaat (1973/1974)

Séminaire Équations aux dérivées partielles (Polytechnique)

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

For a Morse function $f$ on a compact oriented manifold $M$ , we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect to links....