Local holomorphic curves on a bounded symmetric domain in its Harish-Chandra realization exiting at regular points of the boundary

We examine the asymptotic behavior of holomorphic curves on bounded symmetric domains in their Harish-Chandra realizations exiting at smooth points of the boundary, showing that such a holomorphic curve is aysmptotically totally geodesic at a generic point of its boundary and that furthermore the norm of the second fundamental form is bounded by a constant multiple of the Euclidean distance to the boundary.