[Paper Review] The Singular Set of 1-1 Integral Currents

Tristan Rivière, Gang Tian|ArXiv.org|Oct 4, 2003

Geometry and complex manifolds9 references4 citations

TL;DR

This paper establishes that 2-dimensional integral currents which are almost complex cycles in an almost complex manifold admitting a locally compatible symplectic form are smooth J-holomorphic curves except at isolated singular points. Using a geometric measure theory approach inspired by calibrated current theory, the authors prove that such currents arise as pushforwards of smooth J-holomorphic maps from Riemann surfaces with integer multiplicity, extending regularity results beyond the integrable case.

ABSTRACT

We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth surfaces aside from isolated points and therefore are J-holomorphic curves.

Motivation & Objective

• To establish the regularity of 2-dimensional integral currents that are almost complex cycles in almost complex manifolds.
• To show that such currents are realized as smooth J-holomorphic maps from Riemann surfaces, except at isolated points.
• To provide an alternative proof to Almgren's regularity theory, tailored specifically to the almost complex setting.
• To lay groundwork for extending the result to non-locally symplectic almost complex structures.

Proposed method

• The authors work within Geometric Measure Theory, analyzing rectifiable 2-currents with integer multiplicity and closedness under the boundary operator.
• They impose the condition that the approximate tangent planes are invariant under the almost complex structure J, ensuring the current is an almost complex cycle.
• The key assumption is the locally symplectic property: at each point, there exists a local symplectic form compatible with J.
• The proof uses a covering argument with balls of controlled overlap, relying on a Vitali-type covering lemma to control the density of overlapping balls.
• The argument proceeds by contradiction, assuming infinitely many overlapping balls near a point, and derives a contradiction via geometric constraints on radii and centers in the limit.
• The limiting configuration of balls leads to a contradiction with the uniform bounded overlap condition, proving the finitely many overlapping balls condition.

Experimental results

Research questions

• RQ1 Can 2-dimensional integral currents that are almost complex cycles in an almost complex manifold be shown to be smooth J-holomorphic curves except at isolated points?
• RQ2 Under what conditions does the locally symplectic property ensure regularity of such currents?
• RQ3 How can one prove the regularity of almost complex cycles without relying on Almgren's full regularity theory for area-minimizing currents?
• RQ4 Can the proof strategy be adapted to the non-locally symplectic case?

Key findings

• The singular set of a 2-dimensional integral current that is an almost complex cycle in a locally symplectic almost complex manifold consists of isolated points only.
• The current is realized as the pushforward of a smooth J-holomorphic map from a Riemann surface with integer multiplicity, except at isolated points.
• The proof establishes that the number of overlapping balls in a covering of a set is uniformly bounded, which is essential for regularity.
• The result holds under the assumption that the almost complex structure admits a compatible local symplectic form, a condition known to hold in 4-dimensional manifolds.
• The method avoids reliance on Almgren’s full regularity machinery, offering a more direct and adaptable approach.
• The authors indicate that the strategy can be extended to the general case without the locally symplectic assumption via lower-order perturbations.