On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator

Josef Greilhuber; Willi Kepplinger

Home/Archive/arXiv

arXiv · 2312.16939

On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator

Josef Greilhuber·Willi Kepplinger

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace--Beltrami operator has an eigenvalue of a given multiplicity. The starting point of our investigation is the "Strong Arnold Hypothesis" introduced by Colin de Verdi\`ere, which posits that Laplace eigenvalues of higher multiplicity split up under perturbation exactly as eigenvalues of symmetric matrices do. There exists a simple geometric characterization, due to Colin de Verdi\`ere and Besson, of metrics which satisfy the Strong Arnold Hypothesis. Using Besson's characterization, we prove the Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension, and use this to obtain the precise codimension of the set of metrics admitting an eigenvalue of any given multiplicity. Furthermore, we show that the Strong Arnold Hypothesis is satisfied for all metrics admitting eigenvalues of multiplicity at most six, and discuss several examples of metrics violating it.

Explore related subjects

math.DG math.AP math.SP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Josef Greilhuber, Willi Kepplinger. 2023-12-28. On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator. https://arxiv.org/abs/2312.16939

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The $q$-deformed cross-ratio: modular invariants and Coxeter friezes

We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

math.DG↗

arXiv2026

The Cartan-Hadamard conjecture in dimension five

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

math.DG↗

arXiv2026

On static manifolds with boundary admitting a nowhere-vanishing static potential

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

math.DG↗