A local isoperimetric inequality for balls with nonpositive curvature

We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation corresponds to a homothety of the ball. These results establish a sharp local version of the Cartan-Hadamard conjecture.

💡 Research Summary

The paper addresses a local version of the Cartan‑Hadamard isoperimetric conjecture. The classical conjecture asserts that for any domain Ω in a complete simply‑connected manifold with non‑positive sectional curvature, the isoperimetric ratio I(Ω)=|∂Ω|ⁿ/|Ω|ⁿ⁻¹ is at least that of the Euclidean unit ball, with equality only for Euclidean balls. The authors prove that this inequality holds for small C²‑perturbations of the Euclidean metric on the unit ball Bⁿ, and that equality forces the perturbed metric to be a homothety of the Euclidean metric.

The setting is the space M₀(Bⁿ) of smooth metrics g on Bⁿ with non‑positive curvature, equipped with the C²‑norm. The main result (Theorem 1.1) states that there exists ε>0 such that if |g−δ|_{C²}≤ε then I(Bⁿ_g)≥I(Bⁿ_δ), and equality implies g is isometric to the Euclidean metric. The proof proceeds by first showing that for sufficiently small ε the perturbed ball Bⁿ_g is strictly convex and therefore a CAT(0) space. Using normal coordinates at the center, Bⁿ_g can be identified with a star‑shaped domain Ω⊂ℝⁿ whose boundary is the graph of a radial function f_g on the unit sphere S^{n−1}. The function f_g is C¹‑close to the constant 1.

A key technical ingredient is Rauch’s comparison theorem, which yields a pointwise comparison between the Jacobian J_g(rθ) of the exponential map of (Bⁿ,g) and the Jacobian J_k(r) of the model space of constant curvature k≤0 (k=0 gives the Euclidean case). Lemma 3.2 provides a matrix inequality: if two symmetric positive‑definite matrices satisfy A≥B pointwise then det A≥det B and det A·A⁻¹≥det B·B⁻¹, with equality only when A=B. This matrix result is used to translate curvature bounds into Jacobian bounds.

The volume and perimeter of Ω_g are expressed in polar coordinates as integrals involving J_g and the gradient of f_g. By inserting the comparison inequalities, the authors derive a lower bound for the isoperimetric ratio in terms of a one‑parameter family λ(t). They show λ′(t)≥0 when f_g and its gradient are sufficiently close to the Euclidean values, which holds for ε small enough. Consequently λ(1)≥λ(0) and I(Ω_g)≥I(Ω_{δ_k}). Equality forces the auxiliary factor ρ(θ)=J_g(f(θ)θ)/J_k(f(θ)) to be identically one, which by the rigidity part of Rauch’s theorem implies g=δ_k.

The paper also discusses limitations. Note 1.2 points out that without the smallness assumption the inequality fails: there exist negatively curved metrics on B³ with arbitrarily small isoperimetric ratio. Note 1.3 shows that the result does not extend to geodesic balls in hyperbolic space, where the isoperimetric ratio increases with radius.

In summary, the authors establish a sharp local isoperimetric inequality for balls with non‑positive curvature, confirming the Cartan‑Hadamard conjecture in a neighborhood of the Euclidean metric and providing a rigidity statement that characterizes the equality case. The work combines comparison geometry, linear algebraic matrix inequalities, and a variational argument, and it clarifies both the power and the limits of local curvature control in isoperimetric problems.