Constant potentials do not minimize the fundamental gap on convex domains in negatively curved Hadamard manifolds

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.

💡 Research Summary

The paper addresses the second part of the Fundamental Gap Conjecture, which posits that among all convex potentials on a convex domain, the constant potential should minimize the fundamental gap (the difference between the first two Dirichlet eigenvalues). While the conjecture has been proved for Euclidean spaces and partially for hyperbolic space, this work shows that it fails in every negatively curved Hadamard manifold.

The author fixes a complete, simply connected Riemannian manifold X with everywhere negative sectional curvature. For any prescribed diameter D>0, a convex domain Ω⊂X of diameter D is constructed. The domain is built as a thin tubular neighbourhood of a chosen geodesic γ₀, with a “neck” region that becomes arbitrarily narrow as a parameter ε→0. This geometry forces the first two eigenfunctions of the Laplacian to concentrate away from the neck, mimicking a one‑dimensional situation.

A convex potential V is then defined on Ω, depending only on the coordinate t along γ₀, and chosen to be monotone increasing after a certain point. The key variational identity is
∫_Ω V (u₂²−u₁²) dv < 0,
where u₁ and u₂ are the first two Dirichlet eigenfunctions of −Δ on Ω. By the Hellmann–Feynman theorem, the derivative of the gap Γ(Ω; rV) with respect to the scaling parameter r at r=0 equals exactly the left‑hand side of the inequality. Hence, for sufficiently small r>0 the gap with potential rV is strictly smaller than the gap with the constant (zero) potential.

To justify the inequality, the paper develops a sophisticated approximation scheme. In local coordinates (s,t) adapted to the geodesic, the metric is written as g=dt²+g_{ij}(s,t) ds^i ds^j + g_{ti}(s,t) dt ds^i, with g_{ti}(0,t)=0 due to the vanishing second fundamental form at the base point. The author introduces a simplified metric g⁰ obtained by freezing the s‑dependence at s=0, leading to a model Laplacian Δ⁰ that depends only on t. After a rescaling x=δ^{−1/3}(t₀−t), y=s/ε (with δ(ε)→0), the operators become \tildeΔ and \tildeΔ⁰ on a fixed rectangular domain.

The model eigenvalue problem for \tildeΔ⁰ reduces, after further scaling, to a perturbed Airy equation. Lemma 4.1 uses the negative curvature to guarantee positivity of the shape operator along the geodesic, which in turn ensures that the coefficients of the ODE are close to those of the Airy operator. Proposition 4.5 shows that the eigenfunctions of \tildeΔ⁰ can be approximated by separated‑variables expressions built from the Airy eigenfunctions, with an error of order δ^α.

Section 5 then bridges the gap between the true operator \tildeΔ and its model \tildeΔ⁰. Using standard perturbation estimates (Lemma 3.1) and H²‑regularity up to the non‑smooth boundary (Remark 2.6), the first two eigenfunctions of \tildeΔ are shown to be close in L² to those of \tildeΔ⁰. Consequently, the integral ∫_Ω V (u₂²−u₁²) can be evaluated using the explicit Airy asymptotics, yielding a negative value for any monotone increasing V.

The final argument (Section 5.2) applies the Hellmann–Feynman identity to conclude that for the constructed convex domain Ω and convex potential V, the fundamental gap satisfies Γ(Ω; V) < Γ(Ω). Since D>0 was arbitrary, this disproves the conjectured minimality of constant potentials in all negatively curved Hadamard manifolds.

Beyond the specific result, the paper demonstrates a robust method for handling spectral problems on manifolds lacking symmetry. By localizing the domain around a geodesic, approximating the Laplacian with a one‑dimensional model, and exploiting the Airy equation’s well‑understood spectral properties, the author overcomes the obstacle that, unlike hyperbolic space, general negatively curved manifolds do not admit a reduction to ordinary differential equations. This approach opens the door to further investigations of spectral gaps, heat kernel estimates, and related functional inequalities in non‑symmetric geometric settings.