Uniform mass lower bound for hyperbolic-surface eigenfunctions

Statement

Let $M$ be a compact hyperbolic surface and $U\subset M$ a nonempty open set. There is $c_U>0$ such that every $L^2$ -normalized eigenfunction $\psi_k$ of the Laplace–Beltrami operator satisfies

\|\psi_k\mathbf1_U\|_{L^2(M)}\ge c_U.

The constant depends on $U$ and $M$ , but not on the eigenvalue.

Semiclassical interpretation

Writing a high eigenvalue as $h^{-2}$ , the theorem rules out a sequence of eigenfunctions whose mass in $U$ tends to zero as $h\to0$ . Equivalently, every semiclassical measure arising from eigenfunctions has full support.

Role of fractal uncertainty

Boundary data giving incoming and outgoing hyperbolic plane-wave representations would both have to concentrate near the porous set of geodesic endpoints avoiding $U$ . Their oscillatory relation and the fractal uncertainty principle exclude simultaneous concentration.

References

Semyon Dyatlov and Long Jin, “Semiclassical measures on hyperbolic surfaces have full support,” Acta Mathematica 220 (2018), 297–339. DOI record.