Rate of decay for the mass ratio of pseudo-holomorphic integral 2-cycles

TL;DR

This paper establishes a geometric decay rate for the mass ratio of pseudo-holomorphic integral 2-cycles in almost complex manifolds, providing explicit decay exponents based on local density, advancing understanding of their local geometric behavior.

Contribution

The paper introduces a new decay rate result for pseudo-holomorphic cycles using a pseudo algebraic blow-up technique, extending previous methods to arbitrary points in almost complex manifolds.

Findings

Proves a geometric decay rate for the mass ratio.

Provides explicit decay exponents related to local density.

Extends blow-up analysis to arbitrary points in the manifold.

Abstract

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an explicit exponent of decay expressed in terms of the density of the current at the point.