arXiv research

New bounds found for nodal sets on special manifolds.

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

Study conformal invariants from nodal sets on manifolds with boundary.

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 studies eigenfunctions and nodal sets of the Witten-Laplacian.

Eigenfunction maxima inside high-d nodal domains.

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

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

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

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

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 $\sim \frac{1}{\sqrtλ}$. …

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.

Formulas calculate zero-set volume on Riemannian manifolds.

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.

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 goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by …

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

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…

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 study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and…

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…

Study ALE spaces via nodal curves and compactifications.

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

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…

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

Compactness theorem for quasiregular curves proves normality and resolves nodal points.

Compactness theorem for quasiregular maps between manifolds.

The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(λ) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal…

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal…

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidi…

We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $Σ$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to…

New chaos formula simplifies variance calculation for Gaussian nodal volumes.

Study finds solutions to Allen-Cahn equation on manifolds with symmetry.

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $φ_λ$ on a closed Riemannian manifold $(M,g)$. First, in the real-analytic case we present an improvement of the currently best known bounds, due to Mangoubi (\cite{Man1}). Furthermore, using recent re…

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 study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metr…

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

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be boun…

In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a 2-d analogue of this fact: there is a close relationship between the topology of the contact structure on a convex surfac…

New nodal domain theorems for symmetric matrices via signed graphs.

Study how nodal domains change on surfaces under perturbations.

Study on variance of Laplace eigenfunctions on manifolds.

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…

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

Sharp uncertainty principle for nodal sets in singular spaces.

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.