Spectral Turán Problems for Nondegenerate Hypergraphs

Keevash, Lenz and Mubayi developed a general criterion for hypergraph spectral extremal problems in their seminal work (SIAM J. Discrete Math., 2014). Their framework shows that extremal results on the $\alpha$-spectral radius (for $\alpha > 1$) may be deduced from a corresponding hypergraph Turán problem exhibiting stability properties, provided its extremal construction satisfies certain continuity assumptions. In this paper, we establish a spectral stability result for nondegenerate hypergraphs, extending the Keevash-Lenz-Mubayi criterion. Applying this result, we derive two general spectral Turán theorems for hypergraphs with bipartite or multipartite pattern, thereby transforming spectral Turán problems into the corresponding purely combinatorial problems related to degree-stability in nondegenerate $k$-graph families. As applications, we determine the maximum $\alpha$-spectral radius for several classes of hypergraph and characterize the corresponding extremal hypergraphs, such as the expansion of complete graphs, the generalized fans, the cancellative hypergraphs, the generalized triangles, and a special book hypergraph.