Estimates for J-curves as submanifolds

TL;DR

This paper develops analytic tools to study the compactness and regularity of pseudo-holomorphic curves as submanifolds in almost Hermitian manifolds, extending techniques from minimal surface theory.

Contribution

It introduces new estimates and regularity results for J-curves, including an inhomogeneous mean curvature equation and epsilon-regularity, applicable with or without boundary.

Findings

Derived an inhomogeneous mean curvature equation for J-curves

Established an extrinsic monotonicity principle for certain functions

Proved epsilon-regularity for the second fundamental form

Abstract

Here we develop some basic analytic tools to study compactness properties of $J$-curves (i.e. pseudo-holomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves, we establish an extrinsic monotonicity principle for non-negative functions $f$ satisfying $\Delta f\geq -c^2 f$, we show that curves locally parameterized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we establish $\epsilon$-regularity for the square length of their second fundamental forms. These results are all provided for $J$-curves either with or without Lagrangian boundary and hold in almost Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).