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Yakun Xi

Publications and source records attributed to Yakun Xi.

At least 19 recordsLinked to original sources
arXiv2026

Restriction estimates for surfaces with negative curvature in $\mathbb R^3$

In $\mathbb R^3$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q'<\frac{p}{2}$. Building on Demeter--Wu's work for the model hyperbolic paraboloid, we introduce a geometric propagation principle for good lines, which controls the degenerate directions arising from straight-line segments on the surface. This overcomes a key difficulty in the general case, where such directions may vary with the geometry rather than being fixed by the coordinate axes.

math.CA↗
arXiv2026

Polygons and multi-product of eigenfunctions

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j\}_j$, which satisfy $\Delta_g e_j = -\lambda_j^2 e_j$. We study the following inner product of eigenfunctions \[ \langle e_{i_1} e_{i_2} \ldots e_{i_k}, e_{i_{k+1}} \rangle = \int e_{i_1} e_{i_2}\ldots e_{i_k} \overline{e_{i_{k+1}}} \, dV. \] We show that, after a mild averaging in the frequency variables, the main $\ell^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.

math.AP↗
arXiv2025

Kuznecov formulae for fractal measures

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum \[ N_\mu(\lambda):=\sum_{\lambda_j\le \lambda}\Bigl|\int_M e_j\,d\mu\Bigr|^2. \] Assume that $\mu$ admits an averaged $s$-density constant $A_\mu$ with correlation dimension $s\in(0,n)$. We prove that \[N_\mu(\lambda)= (2\pi)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_\mu\,\lambda^{n-s}+ o(\lambda^{n-s})\qquad (\lambda\to\infty). \] The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o(\lambda^{n-s})$ is sharp in the sense that it cannot be improved uniformly to a power-saving error term. This extends the classical Kuznecov formula of Zelditch for smooth submanifold measures to a broad class of singular and fractal measures.

math.AP↗
arXiv2025

Restriction and Kakeya maximal estimates in $\mathbb{R}^4$

By combining the planebrush argument of Katz and Zahl \cite{katz21} with the decoupling-incidence method of Wang and Wu \cite{WangWu2024}, we derive new bounds for the Fourier restriction problem and the Bochner--Riesz problem, extending the range to $p > 2 + \frac{200}{251}$ in $\mathbb{R}^4$. Moreover, leveraging the two-ends Furstenberg estimate in the plane, we also obtain a Kakeya maximal estimate in $\mathbb{R}^4$ at dimension $3.054$.

math.CA↗
arXiv2025

Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications

We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. On manifolds of constant sectional curvature, we introduce a new anisotropic variant of the microlocal Kakeya--Nikodym norm that further enlarges the admissible $p$-range. As a corollary, by combining our results with a recent theorem of Hou, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds. In particular, our method applies to general H\"ormander operators, and we characterize the $L^q \to L^p$ boundedness of H\"ormander operators with positive-definite phase in all dimensions $n\ge3$, thereby fully resolving a question going back to H\"ormander. Further applications include improved $L^q \to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in $\mathbb R^3$.

math.CA↗
arXiv2025

Curved Kakeya sets for generic phases in odd dimensions

We show that for each odd integer $n\ge 3$, there is an open dense subset of H\"ormander phase functions in $\mathbb{R}^n$ for which the associated curved Kakeya sets have Hausdorff dimension at least $\frac{n+1}{2} + d_n$ for some positive $d_n$, thereby exceeding the classical compression threshold. In particular, in $\mathbb{R}^3$, generic H\"ormander phases induce curved Kakeya sets of dimension at least $2 + \tfrac17$. As an application, on a generic three-dimensional Riemannian manifold, a local Nikodym set has Hausdorff dimension at least $2 + \tfrac17$. We achieve these results by generalizing the finite contact order condition from Dai--Gong--Guo--Zhang, originally developed in $\mathbb{R}^3$, to arbitrary dimensions. Our bounds are stronger than those of Dai--Gong--Guo--Zhang even in $\mathbb{R}^3$, since we derive curved Kakeya estimates directly via the polynomial method. Moreover, for H\"ormander-type oscillatory integral operators with positive-definite phases of finite contact order, we obtain quantitative improvements in all odd dimensions over the bounds of Guth--Hickman--Iliopoulou, while in three dimensions our oscillatory integral estimate exactly matches the result of Dai--Gong--Guo--Zhang.

math.CA↗
arXiv2025

Kuznecov remainders and generic metrics

We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [arXiv:2204.13525], we prove that for a Baire-generic class of metrics, the oscillatory second term in the Kuznecov formula can be eliminated, yielding an improved remainder estimate.

math.AP↗
arXiv2025

Sharp spectral-cluster restriction bounds for orthonormal systems

Let $\Sigma$ be a smooth submanifold of a compact Riemannian manifold $M$. We study the restriction to $\Sigma$ of densities generated by $L^2(M)$-orthonormal systems of eigenfunctions whose eigenfrequencies lie in a unit-width spectral cluster. For every $p\geq2$, we establish $L^{p/2}(\Sigma)$ bounds that quantify the gain arising from orthogonality and determine the optimal summability exponent for the coefficient sequence in all dimensions and codimensions. The summability exponent is fully sharp, while the frequency dependence is optimal modulo logarithmic factors. In the codimension-one case when $\dim M\geq3$, the optimal summability exponent exhibits a new, dimension-independent critical point at $p=4$, in addition to the classical critical point $p=\frac{2d}{d-1}$. This phenomenon has no analogue in the corresponding restriction estimates for a single eigenfunction. We also obtain improved, essentially sharp estimates when $\Sigma$ is a curve with nonvanishing geodesic curvature on a Riemannian surface.

math.AP↗
arXiv2025

Curved Kakeya sets and Nikodym problems on manifolds

In this paper, we study curved Kakeya sets associated with phase functions satisfying Bourgain's condition. In particular, we show that the analysis of curved Kakeya sets arising from translation-invariant phase functions under Bourgain's condition, as well as Nikodym sets on manifolds with constant sectional curvature, can be reduced to the study of standard Kakeya sets in Euclidean space. Combined with the recent breakthrough of Wang and Zahl, our work establishes the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature. Moreover, we consider $(d,k)$-Nikodym sets and $(s,t)$-Furstenberg sets on Riemannian manifolds. For manifolds with constant sectional curvature, we prove that these problems can similarly be reduced to their Euclidean counterparts. As a result, the Furstenberg conjecture on two-dimensional surfaces with constant Gaussian curvature follows from the work of Ren and Wang.

math.CA↗
arXiv2024

Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the $L^p(M)$ estimates of Sogge and the $L^p(\gamma)$ restriction bounds of Burq, G\'erard, and Tzvetkov, and are sharp for all $p \geq 2$, up to a $\lambda^\varepsilon$ loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with nonvanishing geodesic curvature. These estimates are closely related to a variable-coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.

math.AP↗
arXiv2024

Hearing the shape of a drum by knocking around

We study a variation of Kac's question, "Can one hear the shape of a drum?" if we allow ourselves access to some additional information. In particular, we allow ourselves to ``hear" the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is ``yes" provided the Laplace-Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary.

math.AP↗
arXiv2024

Cospectral vertices, walk-regular planar graphs and the echolocation problem

We study cospectral vertices on finite graphs in relation to the echolocation problem on Riemannian manifolds. First, We prove a computationally simple criterion to determine whether two vertices are cospectral. Then, we use this criterion in conjunction with a computer search to find minimal examples of various types of graphs on which cospectral but non-similar vertices exist, including minimal walk-regular non-vertex-transitive graphs, which turn out to be non-planar. Moreover, as our main result, we classify all finite 3-connected walk-regular planar graphs, proving that such graphs must be vertex-transitive.

math.CO↗
arXiv2023

Can you hear your location on a manifold?

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point $x$ on the manifold, up to symmetry, from its pointwise counting function \[ N_x(λ) = \sum_{λ_j \leq λ} |e_j(x)|^2, \] where here $Δ_g e_j = -λ_j^2 e_j$ and $e_j$ form an orthonormal basis for $L^2(M)$. This problem has several natural physical interpretations, two of which are acoustic: 1. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echoes and reverberations? 2. If a drum of a known shape is struck at some unknown point, can you determine this point by listening to the quality of the sound the drum produces? The main result of this paper provides an affirmative answer to this question for a generic class of metrics. We also probe the problem with a variety of simple examples, highlighting along the way helpful geometric invariants that can be pulled out of the pointwise counting function $N_x$.

math.AP↗
arXiv2023

Surfaces in which every point sounds the same

We address a maximally structured case of the question, "Can you hear your location on a manifold," posed in arXiv:2304.04659 for dimension $2$. In short, we show that if a compact surface without boundary sounds the same at every point, then the surface has a transitive action by the isometry group. In the process, we show that you can hear your location on Klein bottles and that you can hear the lengths and multiplicities of looping geodesics on compact hyperbolic quotients.

math.AP↗
arXiv2023

Square function estimates and Local smoothing for Fourier Integral Operators

We prove a variable coefficient version of the square function estimate of Guth--Wang--Zhang. By a classical argument of Mockenhaupt--Seeger--Sogge, it implies the full range of sharp local smoothing estimates for $2+1$ dimensional Fourier integral operators satisfying the cinematic curvature condition. In particular, the local smoothing conjecture for wave equations on compact Riemannian surfaces is completely settled.

math.AP↗
arXiv2023

Improved local smoothing estimate for the wave equation in higher dimensions

In this paper, we establish the sharp $k$-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions $n\ge3$. As a byproduct, we also generalize the restriction estimates of Ou--Wang to a broader class of phase functions.

math.AP↗
arXiv2023

A two term Kuznecov sum formula

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(λ) := \sum_{λ_j \leq λ} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} λ^{\operatorname{codim} H} + O(λ^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $Δ_g e_j = -λ_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(λ) \sim C_{H,M} λ^{\operatorname{codim} H} + Q(λ) λ^{\operatorname{codim} H - 1} + o(λ^{\operatorname{codim} H - 1}) $$ where $Q$ is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive $H$ in the normal directions. In work by Canzani, Galkowski, and Toth, they establish (as a corollary to a stronger result involving defect measures) that if the set of recurrent directions of geodesics normal to $H$ has measure zero, then we obtain improved bounds on the individual terms in the sum -- the period integrals. We are able to give a dynamical condition such that $Q$ is uniformly continuous and `$\sim$' can be replaced with `$=$'. This implies improved bounds on period integrals, and this condition is weaker than the recurrent directions having measure zero. Moreover, our result implies improved bounds for period integrals if there is no $L^1$ measure on $SN^*H$ that is invariant under the first return map. This generalizes a theorem of Sogge and Zelditch and of Galkowski.

math.AP↗