  For a general class of hypergraph Tur\'an problems with uniformity $r$, we
investigate the principal eigenvector for the $p$-spectral radius (in the sense
of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a
strong sense that these eigenvectors have close to equal weight on each vertex
(equivalently, showing that the principal ratio is close to $1$). We
investigate the sharpness of our result; it is likely sharp for the Tur\'an
tetrahedron problem.
  In the course of this latter discussion, we establish a lower bound on the
$p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the
graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu,
Cioab\u{a}--Gregory, and Zhang.
  The case $1 < p < r$ of our results leads to some subtleties connected to
Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory
for the $p$-spectral radius. We raise a conjecture about these issues, and
provide some preliminary evidence for our conjecture.
