  A primary goal in this paper is to study the question that asks when a real
analytic submanifold $M$ in ${\mathbb{C}}^{n+1}$ bounds a real analytic (up to
$M$) Levi-flat hypersurface $\hat{M}$ near $p\in M$ such that $\hat{M}$ is
foliated by a family of complex hypersurfaces moving along the normal direction
of $M$ at $p$, and gives the invariant local hull of holomorphy of $M$ near
$p$. This question is equivalent to the holomorphic flattening problem for $M$
near $p$.
  We will give an affirmative answer to above question when $p$ is a real
complex tangent point with at least one elliptic direction (positively curved
direction). We also obtain a formal flattening theorem under the assumption of
one non-parabolic direction.
