arXiv subjects
Publications and source records attributed to Gabriel Beiner.
We show that the ECH capacities of every closed symplectic 4-manifold with a rational symplectic form are infinite. We also give the first examples of symplectic 4-manifolds whose alternative ECH capacities are infinite. We verify the ECH Weyl law holds with bounded subleading asymptotics for the alternative ECH capacities of any closed symplectic 4-manifold with $b_2^+=1$ or any smooth domain therewithin.
This article relates the theory of embedded contact homology (ECH) with the dynamics of Anosov flows. We show that in many cases the ECH capacities of a symplectic 4-manifold are infinite, including cotangent disk bundles over closed oriented surfaces of genus at least two. We prove that ECH obstructs Reeb Anosov and Hamiltonian Anosov flows, addressing the four-dimensional case of a question posed by Herman in 1998. Further, we obtain Floer-theoretic obstructions to a 3-manifold admitting any Anosov flow. As an application, we give new constraints on the existence of embedded Lagrangians of genus at least two in symplectic 4-manifolds. In an appendix, some related results in all dimensions are proved for capacities constructed from rational symplectic field theory.
In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $\psi_n$, with eigenvalues $\lambda_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $ψ_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.