  We introduce a novel curvature flow, the Heterotic-Ricci flow, as the
two-loop renormalization group flow of the Heterotic string common sector and
study its three-dimensional compact solitons. The Heterotic-Ricci flow is a
coupled curvature evolution flow, depending on a non-negative real parameter
$\kappa$, for a complete Riemannian metric and a three-form $H$ on a manifold
$M$. Its most salient feature is that it involves several terms quadratic in
the curvature tensor of a metric connection with skew-symmetric torsion $H$.
When $\kappa = 0$ the Heterotic-Ricci flow reduces to the generalized Ricci
flow and hence it can be understood as a modification of the latter via the
second-order correction prescribed by Heterotic string theory, whereas when
$H=0$ and $\kappa >0$ the Heterotic-Ricci flow reduces to a constrained version
of the RG-2 flow and hence it can be understood as a generalization of the
latter via the introduction of the three-form $H$. Solutions of Heterotic
supergravity with trivial gauge bundle, which we call Heterotic solitons,
define a particular class of three-dimensional solitons for the Heterotic-Ricci
flow and constitute our main object of study. We prove a number of structural
results for three-dimensional Heterotic solitons, obtaining the complete
classification of compact three-dimensional strong Heterotic solitons as
hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a
left-invariant metric. Furthermore, we prove that all Einstein
three-dimensional Heterotic solitons have constant dilaton. In this direction,
we prove that Einstein Heterotic solitons with constant dilaton are rigid and
therefore cannot be deformed into a solution with non-constant dilaton. This
is, to the best of our knowledge, the first rigidity result for compact
supergravity solutions in the literature.
