  We study the homotopical minimal measures for positive definite autonomous
Lagrangian systems. Homotopical minimal measures are action-minimizers in their
homotopy classes, while the classical minimal measures (Mather measures) are
action-minimizers in homology classes. Homotopical minimal measures are much
more general, they are not necessarily homological action-minimizers. However,
some of them can be obtained from the classical ones by lifting them to
finite-fold covering spaces. We apply this idea of finite covering to the
geodesic flows on surfaces of higher genus. Let $(M,G)$ be a compact closed
surface with genus $g>1$, where $G$ is a complete Riemannian metric on $M$.
Consider the positive definite autonomous Lagrangian $L(x,v)=G_x(v,v)$, whose
Lagrangian system $\phi_t: TM\rightarrow TM$ is exactly the complete geodesic
flow on $TM$. We show that for each homotopical minimal ergodic measure $\mu$
that is supported on a nontrivial simple closed periodic trajectory, there is a
finite-fold covering space $M'$ such that each ergodic preimage of $\mu$ on
$TM'$ is a minimal measure in the classic Mather theory for the Lagrangian
system on $TM'$.
