2009/12/22 by Joel W. Fish, Fish, Joel W.
Mathematics · #32Q65 (Primary) #53D99 (Secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AP #math.DG #math.SG #msc:32Q65 #msc:53D99
paper · pdf · doi:10.48550/arxiv.0912.4445
48 Pages. Modifications: Application section added to the introduction; also NSF grant support updated.
arxiv created 2010/05/05 · arxiv updated 2010/05/06
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 Δf≥ -c2 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 ε-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).