arXiv research

Study on nodal components of random band-limited functions on surfaces, finding a universal law.

This paper is devoted to studying the structure of codimension one singular holomorphic foliations on $({\mathbb C}^3,0)$ without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, …

The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we stud…

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4-th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extre…

Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.

Continuing the program of math.SG/0012067 and math.SG/0310450, we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Tau…

Study on optimal partitions and nodal solutions for the Yamabe equation.

Study how nodal domains change on surfaces under perturbations.

For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.

The Lady Maisry ballads afford us a framework within which to segment a storyline into its major components. Segments and as a consequence nodal points are discussed for nine different variants of the Lady Maisry story of a (young) woman being burnt to death by her family, on account of her becoming pregnant by a forei…

We prove that, given any knot $γ$ in a compact 3-manifold M, there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set $u^{-1}(0)$ has a connected component given by $γ$. Higher dimensional analogs of thi…

We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right)$. Hence if $\fra…

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $π{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the…

We show that, in round spheres of dimension $n\geq3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{Σ_1,...,Σ_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $ψ=(ψ_1,...,ψ_N)$ of the Dirac operator such th…

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in…

Let $Σ$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $Σ$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigen…

Eigenfunction maxima inside high-d nodal domains.

New bounds found for nodal sets on special manifolds.

New nodal domain theorems for symmetric matrices via signed graphs.

We consider a Laplace eigenfunction $\varphi_λ$ on a smooth closed Riemannian manifold, that is, satisfying $-Δ\varphi_λ= λ\varphi_λ$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tu…

The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. …

New method models PDEs from noisy, limited data.

The paper studies nodal sets of solutions to parabolic equations, proving finiteness and monotonicity properties.

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.

New estimates for nodal and singular sets of parabolic inequalities.

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator …

A new spinorial heat flow framework studies geometric degeneration on 3-manifolds.

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove fini…

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

Study compares nodal sets of solutions to the Allen-Cahn equation.

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

Formula conjectured for rational cuspidal curves in projective plane.

Study on the nodal set of Dirac equation solutions on manifolds.

Sharp estimate for nodal domains intersecting a ball on a Riemannian manifold.

Study small perturbations on low energy Laplace eigenfunctions.

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a resul…

We prove a natural inequality which implies the known lower bounds for the $(n-1)$-dimensional Hausdorff measure of nodal sets for smooth compact manifolds.

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

Study ALE spaces via nodal curves and compactifications.

The family Blow Up formula is recalled. Certain combinatoric graphs are introduced for the discussion of the counting of nodal curves on an Kahler surface.

This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic …

Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.