arXiv research

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

We consider a closed cohomogeneity one Riemannian manifold $(M^n,g)$ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomog…

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

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

The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.

Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.

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

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain multiplicity results for both positive and nodal solutions. In particular we prove the …

New estimates for nodal and singular sets of parabolic inequalities.

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

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 …

The paper estimates the measure of nodal sets for solutions to a specific type of Schrödinger equation.

We use a modified Bochner technique to derive an inequality relating the nodal set of eigenspinors to eigenvalues of the Dirac operator on closed surfaces. In addition, we apply this technique to solutions of similar spinorial equations.

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 construct solutions of the Cahn-Hilliard equation whose nodal set converges to a given constant mean curvature hypersurface in a Riemannian manifold.

Eigenfunction maxima inside high-d nodal domains.

New bounds found for nodal sets on special manifolds.

Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,π]$, of the form $w" + (h(r)/\sin r)w'+λ(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with …

New nodal domain theorems for symmetric matrices via signed graphs.

Study how nodal domains change on surfaces under perturbations.

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.

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.

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 $\sim \frac{1}{\sqrtλ}$. …

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

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.

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…

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

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…

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.

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

Study small perturbations on low energy Laplace eigenfunctions.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to $g$ and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

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

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_λ: Δ_M \varphi_λ+ λ\varphi_λ= 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $λ$. We show that $H^{n-1}(\{ \varphi_λ=0\}) \leq C λ^α,$ where $α>1/2$ is a constant, whi…

We prove that given a minimal hypersurface $Γ$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $Γ$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2Δu +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets co…

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 …

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.

Study ALE spaces via nodal curves and compactifications.

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…

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 …

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

In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by …

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

New minimal surfaces derived from helicoids.