  In the modified measure formulation the string tension appear as an
additional dynamical degree of freedom and these tensions are not universal,
but rather each string generates its own tension, which can have a different
value for each string. We consider new background field that can couple to
these strings, the tension scalar is capable of changing locally along the
world sheet the value of the tension of the extended object. When many types of
strings probing the same region of space are considered this tension scalar is
constrained by the requirement of quantum conformal invariance. For the case of
two types of strings probing the same region of space with different
dynamically generated tensions, there are two different metrics, associated to
the different strings, that have to satisfy vacuum Einsteins equations and the
consistency of these two equation determines the tension scalar. In a previous
paper we studied solutions that completely avoid singularities, but with
positive and negative tensions strings. In this paper we will consider only
positive tension strings for the cosmological case and for warped space time .
In both of these cases there is a region where the string tensions approach
infinity, in the cosmological case this takes place in the early universe while
in the warped case, for some value of the warping coordinate and since the
Hagedorn temperature is proportional the the string tension, we get this way
string scenarios with no limiting Hagedorn temperature in the early universe
and this opens the possibility of a string cosmology without a Hgedorn phase
transition through all of its history . Similar situation can take place in a
warped space time, for this case, assuming the string have a tendency to avoid
regions with lower Hagedorn temperature, we obtain a mechanism for condensation
of strings into a surface at high temperatures.
