There exists a holomorphic quadratic differential defined on any
H
H
-surface immersed in the homogeneous space
E
(
κ
,
τ
)
{\mathbb {E}(\kappa , \tau )}
given by U. Abresch and H. Rosenberg, called the Abresch–Rosenberg differential. However, there was no Codazzi pair on such an
H
H
-surface associated with the Abresch–Rosenberg differential when
τ
≠
0
\tau \neq 0
. The goal of this paper is to find a geometric Codazzi pair defined on any
H
H
-surface in
E
(
κ
,
τ
)
{\mathbb {E}(\kappa , \tau )}
, when
τ
≠
0
\tau \neq 0
, whose
(
2
,
0
)
(2,0)
-part is the Abresch–Rosenberg differential. We denote such a pair as
(
I
,
I
I
AR
)
(I,II_\textrm {AR})
, were
I
I
is the usual first fundamental form of the surface and
I
I
AR
II_\textrm {AR}
is the Abresch–Rosenberg second fundamental form.
In particular, this allows us to compute a Simons’ type equation for
H
H
-surfaces in
E
(
κ
,
τ
)
{\mathbb {E}(\kappa , \tau )}
. We apply such Simons’ type equation, first, to study the behavior of complete
H
H
-surfaces
Σ
\Sigma
of finite Abresch–Rosenberg total curvature immersed in
E
(
κ
,
τ
)
{\mathbb {E}(\kappa , \tau )}
. Second, we estimate the first eigenvalue of any Schrödinger operator
L
=
Δ
+
V
L= \Delta + V
,
V
V
continuous, defined on such surfaces. Finally, together with the Omori–Yau maximum principle, we classify complete
H
H
-surfaces in
E
(
κ
,
τ
)
{\mathbb {E}(\kappa , \tau )}
,
τ
≠
0
\tau \neq 0
, satisfying a lower bound on
H
H
depending on
κ
\kappa
,
τ
\tau
, and an upper bound on the norm of the traceless
I
I
AR
II_\textrm {AR}
, a gap theorem.