Uniform $L^2$ restriction of toral eigenfunctions to hypersurfaces in dimensions $d \ge 4$ (Bourgain–Rudnick conjecture)

On the standard flat torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$, Bourgain and Rudnick (2009) proved that in dimensions $d=2,3$ every Laplace eigenfunction satisfies a uniform, frequency-independent two-sided $L^2$ bound upon restriction to a real-analytic hypersurface with nowhere-vanishing curvature, and announced an upper bound with exponent $\rho(d) < 1/6$ for $d \ge 4$, explicitly leaving open whether $\rho(d) = 0$ there. The problem: prove or disprove, for $d \ge 4$, the uniform two-sided $L^2$ restriction bound for real-analytic hypersurfaces with nowhere-vanishing curvature, together with the corresponding uniform upper bound for an arbitrary real-analytic hypersurface, and determine whether the exponent $\rho(d)$ can be taken equal to zero and, if not, its optimal value.

Background

Let $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$ be the standard flat $d$-dimensional torus. An eigenfunction of the Laplacian $\Delta$ on $\mathbb{T}^d$ with frequency radius $\lambda$ (Laplace eigenvalue $(2\pi\lambda)^2$) is a trigonometric polynomial

$$\varphi_\lambda(x) = \sum_{n \in \mathbb{Z}^d,\ |n| = \lambda} a_n\, e^{2\pi i n\cdot x},$$

whose frequencies are the lattice points on the sphere of radius $\lambda$, a finite set $E_\lambda = \mathbb{Z}^d \cap \lambda S^{d-1}$ of size roughly $\lambda^{d-2}$. A classical theme in spectral theory, initiated by Burq, Gérard and Tzvetkov, is the behavior of the $L^2$-norm of the restriction of $\varphi_\lambda$ to a fixed submanifold $\Sigma$, measured in the induced surface (Riemannian) measure: on a general compact Riemannian manifold the sharp bound for a hypersurface, i.e. a codimension-one submanifold, is $\|\varphi_\lambda\|_{L^2(\Sigma)} \lesssim \lambda^{1/4}\|\varphi_\lambda\|_{L^2}$, improved to $\lambda^{1/6}$ when $\Sigma$ has nonvanishing curvature (its principal curvatures, the eigenvalues of the second fundamental form, are bounded away from zero); both are saturated on round spheres, so no improvement is possible without arithmetic input.

On the flat torus the frequency set is arithmetically constrained, and the torus is expected to be extremal in the opposite direction. Bourgain and Rudnick proved that for $d = 2, 3$, for every real-analytic hypersurface $\Sigma \subset \mathbb{T}^d$ with nowhere-vanishing curvature, there are constants $0 < c_\Sigma < C_\Sigma < \infty$ such that every eigenfunction of sufficiently large frequency satisfies the uniform two-sided bound

$$c_\Sigma \|\varphi_\lambda\|_{L^2(\mathbb{T}^d)} \le \|\varphi_\lambda\|_{L^2(\Sigma)} \le C_\Sigma \|\varphi_\lambda\|_{L^2(\mathbb{T}^d)}.$$

Their proofs run through the distribution of lattice points on the sphere $\lambda S^{d-1}$: Jarník-type bounds on the number of lattice points in short arcs of the circle (Jarník's theorem: an arc of length at most a constant times $\lambda^{1/3}$ on the circle of radius $\lambda$ contains at most two lattice points), bounds on the number in spherical caps (a spherical cap is the portion of the sphere cut off by a half-space; a cap of radius $r$ carries at most about $\lambda^\epsilon(r^{d-1}/\lambda + r^{d-3})$ points in large dimensions), a mean-square equidistribution of cap counts obtained jointly with Sarnak, and cancellation in bilinear exponential sums over transverse caps. The curvature assumption is necessary for the lower bound: the eigenfunctions $\sin(2\pi n_1 x_1)$ vanish identically on the flat hypersurface $x_1 = 0$. In their 2009 note the authors also announce that for $d \ge 4$ the upper bound holds with some exponent $\rho(d) < 1/6$ (positive curvature), and state explicitly that the question whether $\rho(d) = 0$ for $d \ge 4$ remains open, adding that it “is reasonable to believe” the two-sided theorem “holds in any dimension”; the 2012 full version formulates this as its Conjecture 1.9 and remarks that extending the main theorem to arbitrary dimension “remains unsettled”. In the language of recent work, the upper and lower bounds are the trace inequality and the observability inequality for the surface measure of $\Sigma$: the surface measure of a curved hypersurface is expected to be a measure against which all toral eigenfunctions are uniformly comparable to their ambient $L^2$ mass. The question is the flat-torus, high-dimensional counterpart of quantum ergodic restriction phenomena (quantum ergodicity is the equidistribution of almost all eigenfunctions on a manifold whose classical dynamics is chaotic; its restriction variants ask whether that equidistribution persists for eigenfunction mass on a submanifold), and its resolution is expected to hinge on new equidistribution results for lattice points in small caps on $\lambda S^{d-1}$ in dimensions $d \ge 4$.

Problem Statement

Fix a dimension $d \ge 4$ and let $\Sigma \subset \mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$ be a real-analytic hypersurface (a compact embedded codimension-one submanifold), equipped with its induced surface measure. Consider Laplace eigenfunctions of frequency radius $\lambda$, i.e. trigonometric polynomials

$$\varphi_\lambda(x) = \sum_{n \in \mathbb{Z}^d,\ |n| = \lambda} a_n e^{2\pi i n \cdot x},$$

with Laplace eigenvalue $(2\pi\lambda)^2$.

(a) (Upper bound; trace inequality.) Does there exist a constant $C_\Sigma < \infty$ such that every eigenfunction $\varphi_\lambda$ satisfies

$$\|\varphi_\lambda\|_{L^2(\Sigma)} \le C_\Sigma \, \|\varphi_\lambda\|_{L^2(\mathbb{T}^d)}\ ?$$

(b) (Lower bound; observability.) If $\Sigma$ has nowhere-vanishing curvature, do there exist $c_\Sigma > 0$ and $\Lambda_\Sigma < \infty$ such that every eigenfunction with $\lambda > \Lambda_\Sigma$ satisfies

$$\|\varphi_\lambda\|_{L^2(\Sigma)} \ge c_\Sigma \, \|\varphi_\lambda\|_{L^2(\mathbb{T}^d)}\ ?$$

This is Conjecture 1.9 of Bourgain–Rudnick (GAFA 2012) restricted to its open cases $d \ge 4$ (the two-sided statement is proved for $d = 2, 3$). The curvature hypothesis is required only in (b), while (a) is asked for an arbitrary real-analytic hypersurface. The question whether the exponent $\rho(d)$ in Theorem 1.2 of their 2009 note — which announces $\|\varphi_\lambda\|_{L^2(\Sigma)} \le c(\Sigma)\lambda^{\rho(d)}\|\varphi_\lambda\|_{L^2(\mathbb{T}^d)}$ with some $\rho(d) < 1/6$ for smooth hypersurfaces of positive curvature when $d \ge 4$ — can be taken equal to zero concerns the curved-class upper bound, and does not by itself settle the curvature-free case of (a) or the lower bound (b). Answers may establish or refute either direction separately: a refutation must exhibit, for one fixed admissible $\Sigma$, an infinite family of eigenfunctions whose restricted mass ratio $\|\varphi_\lambda\|_{L^2(\Sigma)}^2/\|\varphi_\lambda\|_{L^2(\mathbb{T}^d)}^2$ is unbounded (refuting (a)) or tends to zero (refuting (b)); a single finite example cannot refute these uniform statements. If a uniform bound fails in some dimension, the problem includes determining the optimal exponent $\rho(d)$ there.

References

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