Related papers: Complex zeros of real ergodic eigenfunctions

Let $(M,g,J)$ be a closed K\"ahler manifold with negative sectional curvature and complex dimension $m := \dim_{\mathbb{C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the…

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured…

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds.…

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of…

In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measures with finite quadratic moment on a separable Hilbert space $H$. We will show that…

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^{\alpha} f(n) : = \sum_{m \neq 0} \frac{e^{2 \pi i\alpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $\alpha \in…

Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold are computed to investigate the stability of the tubes. The geodesic equations are used to show that in the case of thick tubes, the curvature of planar (Frenet…

A pseudo-Anosov surface automorphism $\phi$ has associated to it an algebraic unit $\lambda_\phi$ called the dilatation of $\phi$. It is known that in many cases $\lambda_\phi$ appears as the spectral radius of a Perron-Frobenius matrix…

We study the distribution of the common zero sets of $m$-tuples of holomorphic sections of powers of $m$ singular Hermitian pseudo-effective line bundles on a compact K\"ahler manifold. As an application, we obtain sufficient conditions…

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally…

The Quantum Unique Ergodicity (QUE) conjecture of Rudnick-Sarnak is that every eigenfunction phi_n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E_n ->…

A complex symplectic structure on a Lie algebra $\lie h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $\Omega$ of the $(2,0)$-form. Suppose that $\lie h$ is a…

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an…

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed…

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the…

We study the distribution of the eigenvalue condition numbers $\kappa_i=\sqrt{ (\mathbf{l}_i^* \mathbf{l}_i)(\mathbf{r}_i^* \mathbf{r}_i)}$ associated with real eigenvalues $\lambda_i$ of partially asymmetric $N\times N$ random matrices…

This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined…

Let $\Omega\subset\mathbb{R}^d$ be any open set. We consider solutions of $H\psi_\lambda=\lambda \psi_\lambda$, $\lambda\in\mathbb{C}$, where $H$ is an $m$th order complex constant-coefficient elliptic partial differential operator. We…

Let $(M,g)$ be a closed Riemannian manifold and $\sigma$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for…

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an…