Escape of mass for finite-area hyperbolic surfaces

Let M be a noncompact finite-area hyperbolic surface. We give a lower bound for the L² norm of eigenfunctions and Eisenstein series in a compact part of the surface. This extends a result of Soundararajan [arXiv:0901.4060] to the nonarithmetic setting. The crucial components of the proof are a long-time Egorov's theorem designed for this setting, a nonlinear uncertainty principle of Cohen--Dyatlov [arXiv:2609.30261], and an application of Weyl's method for exponential sums.