Geometry of Monge-Ampère structures (Q2275086)

This work is a concise, exhaustive exposition of the main topics involved in the study of the geometric aspects of Monge-Ampère (MA) equations and operators. Such a subject was treated during a minicourse at the Summer school Wisla-18 ``Nonlinear PDEs, their geometry and applications''. These notes are aimed at graduate students and mathematicians with a background in differential geometry and in geometry of nonlinear PDEs. The author presents definitions and examples of various structures that can be considered on an even-dimensional manifold, namely complex, para-complex, complex symplectic, hypercomplex, hyper-para-complex ones. Generalized complex structures on a real vector space and their link with Hitchin pairs of bivectors are examined. Let \(R\) be the recursion operator associated with two symplectic 1-forms \(\omega,\eta\). The author explains the interrelation between \((\omega,\eta)\) and a symplectic pair (holomorphic symplectic form) in the case \(R^2= \mathrm{Id}\), \(R\neq \pm \mathrm{Id}\) (\(R^2= - \mathrm{Id}\)). He also discusses the geometries defined by a triplet of symplectic forms whose recursion operators \(R_i\) satisfy \({R_i}^2= \pm \mathrm{Id}\), \(R_i \neq \pm \mathrm{Id}\). Following an idea of Lychagin, a geometric approach to the study of MA equations can be developed applying the properties of suitable differential forms on the cotangent space \(T^*M\) of a manifold \(M\). Let \(\omega\) be the standard symplectic form on \(T^*M\). One defines a one-to-one correspondence between MA equations on \(M\) and conformal classes of \(\omega\)-effective forms. The classical problem of local equivalence for MA equations can be regarded as a problem of geometric invariant theory. After recalling the concept of symplectically equivalence between two MA equations, the author studies the action of the group \(\mathrm{Sp}(n,R)\) on the space of primitive \(n\)-forms on the Euclidean space \(\mathbb{R}^{2n}\). If \(n=2,3\), this action has a finite number of orbits and the classification of symplectic MA equations with constant coefficients is obtained. A similar result cannot be proved in the case \(n=4\), since the action is not discrete. Moreover, the author considers a \(2m\)-dimensional manifold endowed with a pair of forms (\(\omega,\alpha\)), \(\omega\) being a symplectic 2-form and \(\alpha\) an \(\omega\)-effective form. If \(m=2\), with any non-degenerate MA structure (\(\omega,\alpha\)) is associated an almost hypercomplex (almost hyper-para-complex) structure, for \(\operatorname{pf}\alpha>0\) (\(\operatorname{pf}\alpha<0\)), \(\operatorname{pf}\alpha\) denoting the Pfaffian of \(\alpha\). A method that allows to extend such results to \(4m+2\)-dimensional manifolds endowed with an MA structure is explained. One defines the concept of Hitchin decomposable MA-structure and proves that any structure of this type determines an almost-Kähler (para-Kähler) structure. Explicit examples and a classification theorem are obtained. Moreover, any Hitchin decomposable MA structure \(\omega,\alpha\) with \(\alpha\) closed determines a nearly Calabi-Yau (nearly para Calabi-Yau) structure. The author gives the explicit expression of the generalized almost Calabi-Yau structures that are associated with classical MA equations. Finally, a new approach to modelling stably stratified geophysical flows is explained. In particular, one expresses the MA structure and the corresponding generalized complex structure that are associated with the 2d Dritschel-Vindez equation. Each section of this formative work ends with the subsection ``Notes and further readings''. This stimulates the reader to go deep into the subject, consulting the wide list of references. For the entire collection see [Zbl 1417.35004].