Principal eigenvectors and principal ratios in hypergraph Tur\'an problems

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.