Let
Σ
\Sigma
be a smooth Riemannian manifold,
Γ
⊂
Σ
\Gamma \subset \Sigma
a smooth closed oriented submanifold of codimension higher than
2
2
and
T
T
an integral area-minimizing current in
Σ
\Sigma
which bounds
Γ
\Gamma
. We prove that the set of regular points of
T
T
at the boundary is dense in
Γ
\Gamma
. Prior to our theorem the existence of any regular point was not known, except for some special choice of
Σ
\Sigma
and
Γ
\Gamma
. As a corollary of our theorem
we answer to a question in Almgren’s Almgren’s big regularity paper from 2000 showing that, if
Γ
\Gamma
is connected, then
T
T
has at least one point
p
p
of multiplicity
1
2
\frac {1}{2}
, namely there is a neighborhood of the point
p
p
where
T
T
is a classical submanifold with boundary
Γ
\Gamma
;
we generalize Almgren’s connectivity theorem showing that the support of
T
T
is always connected if
Γ
\Gamma
is connected;
we conclude a structural result on
T
T
when
Γ
\Gamma
consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when
Σ
=
R
m
+
1
\Sigma = \mathbb R^{m+1}
and
T
T
is
m
m
-dimensional.