Let
H
\mathbf {H}
be the mean curvature vector of an
n
n
-dimensional submanifold
M
n
M^n
in a Riemannan manifold. The critical submanifolds of the total mean curvature functional
H
=
∫
|
H
|
n
\mathcal H=\int |\mathbf {H}|^n
are called
H
\mathcal H
-submanifolds. In this note, we will prove that compact Legendrian
H
\mathcal H
-surfaces in the unit sphere
S
5
\mathbb {S}^{5}
are minimal. We also investigate the first eigenvalue of the Schrödinger operator
L
=
−
Δ
−
q
L=-\Delta -q
on
M
2
M^2
and
M
3
M^3
, where
q
q
is some potential function, and obtain a sharp estimate for the first eigenvalue of
L
L
.