The deformation space
C
(
Σ
)
\mathfrak {C}(\Sigma )
of convex
R
P
2
\mathbb {R}{{\mathbf {P}}^2}
-structures on a closed surface
Σ
\Sigma
with
χ
(
Σ
)
>
0
\chi (\Sigma ) > 0
is closed in the space
Hom
(
π
,
SL
(
3
,
R
)
)
/
SL
(
3
,
R
)
\operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R})
of equivalence classes of representations
π
1
(
Σ
)
→
SL
(
3
,
R
)
{\pi _1}(\Sigma ) \to \operatorname {SL} (3,\mathbb {R})
. Using this fact, we prove Hitchin’s conjecture that the contractible "Teichmüller component" (Lie groups and Teichmüller space, preprint) of
Hom
(
π
,
SL
(
3
,
R
)
)
/
SL
(
3
,
R
)
\operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R})
precisely equals
C
(
Σ
)
\mathfrak {C}(\Sigma )
.