In this paper, we study the local biholomorphic property of a real
n
n
-manifold
M
⊂
C
n
M\subset \mathbf C^n
near an elliptic complex tangent point
p
∈
M
p\in M
. In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy
M
~
\widetilde {M}
of
M
M
near
p
p
, first considered in a paper of Bishop. When
M
M
is a
C
∞
C^{\infty }
-smooth submanifold, using a result established by Kenig-Webster, we show that near
p
p
,
M
~
\widetilde {M}
is a smooth Levi-flat
(
n
+
1
)
(n+1)
-manifold with a neighborhood of
p
p
in
M
M
as part of its
C
∞
C^{\infty }
boundary. Moreover, near
p
p
,
M
~
\widetilde {M}
is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to
M
M
. This result was proved in the previous work of Kenig-Webster when
n
=
2
n=2
. When
M
M
is real analytic, we show that
M
~
\widetilde {M}
is real analytic with a neighborhood of
p
p
in
M
M
as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When
n
=
2
n=2
or when
n
>
2
n>2
but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author’s joint work with Krantz.