  We study the magnetic field-dependence of the thermal conductivity due to
magnetic excitations in frustrated spin-1/2 Heisenberg chains. Near the
saturation field, the system is described by a dilute gas of weakly-interacting
fermions (free-fermion fixed point). We show that in this regime the thermal
conductivity exhibits a non-monotonic behavior as a function of the ratio
$\alpha= J_2/J_1$ between second and first nearest-neighbor antiferromagnetic
exchange interactions. This result is a direct consequence of the splitting of
the single-particle dispersion minimum into two minima that takes place at the
Lifshitz point $\alpha=1/4$. Upon increasing $\alpha$ from zero, the inverse
mass vanishes at $\alpha=1/4$ and it increases monotonically from zero for
$\alpha \geq 1/4$. By deriving an effective low-energy theory of the dilute gas
of fermions, we demonstrate that the Drude weight $K_{\rm th}$ of the thermal
conductivity exhibits a similar dependence on $\alpha$ near the saturation
field. Moreover, this theory predicts a transition between a two-component
Tomonaga-Luttinger liquid and a vector-chiral phase at a critical value
$\alpha=\alpha_c$ that agrees very well with previous density matrix
renormalization group results. We also show that the resulting curve $K_{\rm
th}(\alpha)$ is in excellent agreement with exact diagonalization (ED) results.
Our ED results also show that $K_{\rm th}(\alpha)$ has a pronounced minimum at
$\alpha\simeq 0.7$ and it decreases for sufficiently large $\alpha$ at lower
magnetic field values. We also demonstrate that the thermal conductivity is
significantly affected by the presence of magnetothermal coupling.
