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The singular set of 1-1 integral currents. (English) Zbl 1182.32010

Let \((M^{2p},J)\) be an almost complex manifold and consider \(k\in \mathbb{N};\;k\leq p\). A \(2k\)-current \(C\) in \((M^{2p},J)\) is an almost complex integral cycle if it fulfills the conditions of: (i) rectifiability; (ii) closedness; (iii) invariance under the almost complex structure \(J\).
The authors study the regularity of such a cycle. They show that \(2\)-dimensional integer multiplicity \(2\)-dimensional rectifiable currents which are almost complex cycles in an almost complex manifold admitting locally a compatible positive symplectic form are smooth surfaces aside from isolated points and therefore are \(J\)-holomorphic curves.

MSC:

32Q65 Pseudoholomorphic curves
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
32Q60 Almost complex manifolds
49Q05 Minimal surfaces and optimization
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