Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds. (English) Zbl 1232.53066

Chursin, Mykhaylo; Schäfer, Lars; Smoczyk, Knut

Calc. Var. Partial Differ. Equ. 41, No. 1-2, 111-125 (2011).

The authors seek families \(F:M\times[0,T)\to N\) of space-like Lagrangian immersions from a manifold \(M\) into an almost para-Kähler manifold that vary according to the (generalized) mean curvature flow \(F_t(p,t)=\vec H(p,t)\), where \(\vec H\) is a generalized mean curvature vector field on a submanifold of half the ambient dimension, that differs from the usual definition by (lower order) terms that are determined by the para-complex structure.
As their main results, the authors show the existence of a maximal (possibly infinite) time so that the investigated flow admits solutions in the class of semi-dimensional space-like submanifolds in an almost para-Kähler manifold with a metric para-complex connection (Theorem 1), and that the flow preserves the Lagrangian condition if the ambient geometry is endowed with a \(\ast\)-Einstein para-complex connection (Theorem 2).
The paper contains a detailed discussion of the geometric realm of almost para-Kähler manifolds and their (semi-dimensional) submanifolds.

Reviewer: Udo Hertrich-Jeromin (Bath)

MSC:

53D12	Lagrangian submanifolds; Maslov index
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

Lagrangian submanifold; mean curvature flow; Kähler manifold

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