  We continue the development of the effective covariant methods for
calculating the heat kernel and the one-loop effective action in quantum field
theory and quantum gravity. The status of the low-energy approximation in
quantum gauge theories and quantum gravity is discussed in detail on the basis
of analyzing the local Schwinger - De Witt expansion. It is argued that the
low-energy limit, when defined in a covariant way, should be related to
background fields with covariantly constant curvature, gauge field strength and
potential. Some new approaches for calculating the low-energy heat kernel
assuming a covariantly constant background are proposed. The one-loop
low-energy effective action in Yang-Mills theory in flat space with arbitrary
compact simple gauge group and arbitrary matter on a covariantly constant
background is calculated. The stability problem of the chromomagnetic
(Savvidy-type) vacuum is analyzed. It is shown, that this type of vacuum
structure can be stable only in the case when more than one background
chromomagnetic fields are present and the values of these fields differ not
greatly from each other.This is possible only in space-times of dimension not
less than five $d\geq 5$.
