Let \(\Sigma\) be a hypersurface in the standard flat \(d\)-dimensional torus \(T^d\). Let \(\varphi_\lambda\) be an eigenfunction of the Laplacian on \(T^d\). The authors establish uniform upper and lower bounds on the restriction of the eigenfunctions of the Laplacian to smooth hypersurfaces with non-vanishing curvature by showing:
Theorem 1.1. Let \(d=2,3\) and let \(\Sigma\subset T^d\) be a smooth hypersurface with non-zero curvature. There are constants \(0<c<C<\infty\) and \(\Lambda>0\), all depending on \(\Sigma\) so that all eigenfunctions \(\varphi_\lambda\) of the Laplacian on \(T^d\) with \(\lambda>\Lambda\) satisfy \(c\|\varphi_\lambda\|_2\leq\|\varphi_\lambda\|_{L^2\Sigma}\leq C\|\varphi_\lambda\|_2\).
The curvature assumption is clearly necessary for the lower bound as the eigenfunctions \(\varphi_n(x)=\sin(2\pi n_1x_1)\) all vanish on the toroidal hypersurface \(x_1=0\). This lower bound shows a curved hypersurface can not be contained in the nodal set of eigenfunctions with arbitrarily large eigenvalues. In arbitrary dimensions, one has:
Theorem 1.2. For all \(d\geq4\), there is \(p(d)<1/6\) so that if \(\varphi_\lambda\) is an eigenfunction of the Laplacian on \(T^d\) and if \(\Sigma\subset T^d\) is a smooth compact hypersurface with positive curvature, then \(\|\varphi_\lambda\|_{L^2(\Sigma)}\ll\lambda^{p(d)}\|\varphi\|_2\).