  The Landau-Wilson field theory with $O(n)\times O(m)$ symmetry which
describes the critical thermodynamics of frustrated spin systems with
noncollinear and noncoplanar ordering is analyzed in $4 - \varepsilon$
dimensions within the minimal subtraction scheme in the six-loop approximation.
The $\varepsilon$ expansions for marginal dimensionalities of the order
parameter $n^H(m,4-\varepsilon)$, $n^-(m,4-\varepsilon)$,
$n^+(m,4-\varepsilon)$ separating different regimes of critical behavior are
extended up to $\varepsilon^5$ terms. Concrete series with coefficients in
decimals are presented for $m=\{2, \dots, 6\}$. The \textit{diagram of
stability} of nontrivial fixed points, including the chiral one, in $(m,n)$
plane is constructed by means of summing up of corresponding $\varepsilon$
expansions using various resummation techniques. Numerical estimates of the
chiral critical exponents for several couples $\{m,n\}$ are also found.
Comparative analysis of our results with their counterparts obtained earlier
within the lower-order approximations and by means of alternative approaches is
performed. It is confirmed, in particular, that in physically interesting cases
$n=2, m=2$ and $n=2, m=3$ phase transitions into chiral phases should be
first-order.
