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**Minimizing area among Lagrangian surfaces: the mapping problem.**
*(English)*
Zbl 1052.53056

In this exciting paper the authors study symplectic \(4\)-manifolds \(N\). Denote by \(\omega\), \(g\) and \(J\) respectively its symplectic form, its compatible metric and its almost complex structure \(J\).

First the notion of a Lagrangian homology class is introduced, i.e. a class which can be represented by an integral Lagrangian cycle. Various characterisations are obtained, including the fact that such a class can be represented by a piecewise \(C^1\)-Lagrangian map which is an immersion except at finitely many points. Of course, a condition using the first Chern class is given, which if satisfied shows that the Lagrangian map is actually an immersion everywhere.

Next the notion of \(W^{1,2}(\Sigma,N)\) is introduced. Roughly speaking this is the space of maps with square integrable first derivatives. A map is called weakly Lagrangian if the pull-back of the symplectic form vanishes almost everywhere. For weakly conformal, locally exact Lagrangian maps the first variation of the area in a suitable functional space is investigated.

It is also shown, that under a suitable condition, a sequence of minimizing Lagrangian maps has a subsequence which converges strongly to a minimizing Lagrangian map. The above result is then used in order to obtain a global regularity theorem for minimizing Lagrangian maps.

The next sections of the paper then deal respectively with the existence of minimisers in each homotopy class of maps, the second variation formula and a complete description of all stationary Lagrangian cones in \(\mathbb R^4\). The second variation formula is used in order to study their stability and minimisation properties.

First the notion of a Lagrangian homology class is introduced, i.e. a class which can be represented by an integral Lagrangian cycle. Various characterisations are obtained, including the fact that such a class can be represented by a piecewise \(C^1\)-Lagrangian map which is an immersion except at finitely many points. Of course, a condition using the first Chern class is given, which if satisfied shows that the Lagrangian map is actually an immersion everywhere.

Next the notion of \(W^{1,2}(\Sigma,N)\) is introduced. Roughly speaking this is the space of maps with square integrable first derivatives. A map is called weakly Lagrangian if the pull-back of the symplectic form vanishes almost everywhere. For weakly conformal, locally exact Lagrangian maps the first variation of the area in a suitable functional space is investigated.

It is also shown, that under a suitable condition, a sequence of minimizing Lagrangian maps has a subsequence which converges strongly to a minimizing Lagrangian map. The above result is then used in order to obtain a global regularity theorem for minimizing Lagrangian maps.

The next sections of the paper then deal respectively with the existence of minimisers in each homotopy class of maps, the second variation formula and a complete description of all stationary Lagrangian cones in \(\mathbb R^4\). The second variation formula is used in order to study their stability and minimisation properties.

Reviewer: Luc Vrancken (Valenciennes)