Broadly-pluriminimal submanifolds of Kähler-Einstein manifolds (Q2726681)

This article concerns immersions \(F:M\to N\) of a real \(2n\)-dimensional manifold \(M\) in a complex \(2n\)-dimensional Kähler--Einstein manifold \(N\) with Kähler form \(\omega\). The authors introduce the notion of a broadly-pluriminimal immersion, and deduce some properties of such immersions when the scalar curvature \(R\) of \(N\) is negative. A broadly-pluriminimal immersion has slightly weaker properties than a minimal pluriharmonic immersion. Recall that a map is pluriharmonic if the restriction of the map to any complex curve in \(M\) is harmonic: naturally, this requires that \(M\) possesses some complex structure. Such a complex structure can almost be obtained from the pullback \(F^*\omega\) of the Kähler form by polar decomposition, that is, using the induced metric on \(M\) identify \(F^*\omega\) with an endomorphism of the tangent bundle \(TM\) and take its polar decomposition: when \(F^*\omega\) has maximal rank one of the factors in this decomposition produces a complex structure on \(TM\).NEWLINENEWLINENEWLINEThe definition of broadly-pluriminimal is designed to cater for the possibility that \(F^*\omega\) is not of maximal rank. In that case the polar decomposition only equips a subbundle \(K^\perp\) of \(TM\) with a complex structure (\(K\) is the kernel bundle of \(F^*\omega\)). To obtain a complex structure on \(TM\) requires making a choice of complex structure for the complementary subbundle \(K\). The immersion is broadly-pluriminimal if it is minimal and pluriharmonic with respect to any complex structure obtained in this fashion (in fact, the definition allows for the rank of \(F^*\omega\) to vary over \(M\) and takes different local complex structures on those open subsets where the rank is constant). NEWLINENEWLINENEWLINEThe authors prove the following results for broadly-pluriminimal immersions with Kähler--Einstein target. If \(M\) is compact, \(n\geq 2\) and \(R<0\) then: (i) \(F\) has either complex or Lagrangian directions; (ii) if \(n=2\), \(M\) is oriented and \(F\) has no complex directions then \(F\) is Lagrangian. Further, if \(F\) has constant Kähler angles, is not Lagrangian and has no complex directions, then \(R=0\).