Constant potentials do not minimize the fundamental gap on convex domains in negatively curved Hadamard manifolds
Frieder Jäckel
TL;DR
The paper proves that on negatively curved Hadamard manifolds, the fundamental gap Γ(Ω;V) for the Dirichlet problem of −Δ+V can be strictly smaller than the gap for the constant potential, thereby refuting the second part of the Fundamental Gap Conjecture in this broad setting. The authors achieve this by concentrating convex domains Ω along a fixed geodesic and constructing a convex potential V that increases along that axis, then applying Hellmann–Feynman theory to reduce the problem to a model PDE. A key innovation is a PDE-based reduction to a model ODE with Airy-type behavior, together with a careful separation-of-variables argument that approximates Δ by a cylindrical model Δ_∘ and its eigenfunctions via Airy functions. The final step shows that the variational derivative of Γ under a convex perturbation V = rP is negative, giving Γ(Ω;V) < Γ(Ω) for small r and thus establishing the failure of the conjecture in this setting. This extends prior results in hyperbolic space to the wider class of negatively curved Hadamard manifolds and highlights the role of curvature in spectral optimization problems.
Abstract
We show that for every negatively curved Hadamard manifold $X$ and every $D > 0$ there exists a convex domain $Ω\subseteq X$ with diameter $D$ and a convex potential $V$ on $Ω$ such that the fundamental gap of the operator $-Δ+V$ is strictly smaller than the fundamental gap of $-Δ$. This shows that the second part of the fundamental gap conjecture is wrong in every negatively curved manifold. This is significantly harder than in the previously known case of hyperbolic space because, due to the lack of symmetry, one has to study a true PDE, and not just an ODE.