Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds

TL;DR

This work reframes nodal sets of Laplace-type eigenfunctions on compact 2D manifolds as embedded metric graphs and derives Euler-characteristic–driven bounds on the number of nodal-graph vertices (critical points) and on the sum of vanishing orders. By combining Hartman–Wintner regularity with graph-theoretic Euler inequalities, the authors bound these nodal features in terms of the nodal count $μ(u)$ and the surface topology $χ(M)$, with sharper bounds when the nodal set is cellular. The results specialize to eigenfunctions, yielding bounds dependent on the eigenvalue index $k$ and Weyl-law scaling, and they are demonstrated to be sharp on spheres and tori via explicit examples. Extensions to planar domains with boundary and finite contour counts are discussed, along with constructions showing the necessity of finiteness for the bounds. Overall, the paper provides a global, topological perspective on the extremal behavior of eigenfunctions through nodal-graph theory.

Abstract

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, two-dmensional Riemannian manifolds, that is on Riemannian surfaces, as an embedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the $k$-th eigenfunction and the sum of vanishing orders at critical points in terms of $k$ and the Euler-Poincaré characteristic of the surface.

Figures (6)

• Figure 2.1: Closed orientable surfaces. The number $g$ denotes the genus.
• Figure 2.2: Connecting different connected components of the nodal set as in Proposition \ref{['prop:improvedeulersinequality']}. The set $D$ will remain connected.
• Figure 2.3: Removing one handle $U_j$, which is attached at the dashed line, by adding four vertices and four edges (dotted line)
• Figure 4.1: Different nodal sets of spherical harmonics. The left is an example for which the optimal bounds for the number of critical points. The right spherical hamonics function does not satisfy the optimal bound for the number of critical points, however the vanishing orders satisfy the optimal bound.
• Figure 4.2: Nodal set of the eigenfunction $u(x_1,x_2) = \cos(2 x_1) \cos(x_2)$ on the flat torus (opposides side of the square are identified) with eight critical points and eight nodal domains, illustrating sharpness of the upper bound of Theorem \ref{['thm:main_1']}.

Theorems & Definitions (42)

• Definition 2.1: Nodal set and nodal domains
• Proposition 2.2: Courant's nodal domain theorem
• Definition 2.3
• Remark 2.4
• Proposition 2.5: Hartman-Wintner theorem on manifolds, cf. Cheng-76 for the case without boundary and $m = 0$
• Corollary 2.6
• Definition 2.7
• Definition 2.8
• Remark 2.9
• Remark 2.10