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Let $(\pi,\sigma)$ traverse a sequence of pairs of cuspidal automorphicrepresentations of a unitary Gan--Gross--Prasad pair $({\rm U}_{n+1},{\rmU}_n)$ over a number field, with ${\rm U}_n$ anisotropic. We assume that atsome distinguished archimedean place, the pair stays away from the conductordropping locus, while at every other place, the pair has bounded ramificationand satisfies certain local conditions (in particular, temperedness). We provethat the subconvex bound \[ L(\pi \times \sigma,1/2) \ll C(\pi \times\sigma)^{1/4 - \delta} \] holds for any fixed \[ \delta < \frac{1}{8 n^5 + 28n^4 + 42 n^3 + 36 n^2 + 14 n}. \] Among other ingredients, the proof employs arefinement of the microlocal calculus for Lie group representations developedwith A. Venkatesh and an observation of S. Marshall concerning the geometricside of the relative trace formula.

Spectral aspect subconvex bounds for ${\rm U}_{n+1} \times {\rm U}_{n}$