The generalized Cartan-Hadamard conjecture

Ranked by how close their title and abstract are to this paper’s. Select some to read them together.

Closest papers

1. Very close Another approach to the problem Cited by this paper Not relevant

   The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds

   Shi-Bing Chen, M. Ghomi, Peng Wang · 2026 · cited 3 times · Not summarized yet

   The paper proves the Cartan-Hadamard isoperimetric inequality using sharp CMC hypersurface estimates and geodesic-chord integrals.

   Abstract

   We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $n$-manifolds of nonpositive sectional curvature, $3\leq n\leq9$, which establishes the Cartan-Hadamard conjecture in these dimensions. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, together with estimates for Jacobi fields along geodesic chords of the boundary. The weights in these integrals depend on the length of the chord and its angles with the boundary, and are chosen dimension by dimension, with a Green function pole. The inequality persists for boundaries of isoperimetric regions trappe…

2. Very close Another approach to the problem Cited by this paper Not relevant

   The Cartan-Hadamard conjecture in dimension five

   Shi-Bing Chen, M. Ghomi, Peng Wang · 2026 · cited 5 times · Not summarized yet

   The paper proves the dimension-five Cartan-Hadamard inequality using CMC hypersurface estimates and an isoperimetric-profile argument.

   Abstract

   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 ne…

3. Very close Same question Cited by this paper Used in depth by this paper Not relevant

   A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound

   M. Agnoletto, M. D. Da Silva, S. Nardulli et al. · 2026 · cited 1 time · Not summarized yet

   The paper studies the generalized Cartan-Hadamard isoperimetric inequality, establishing it for small volumes under a lower Ricci curvature bound.

   Abstract

   We prove the generalized Cartan-Hadamard conjecture, also known as the Aubin conjecture, in the small volume regime under a lower Ricci curvature bound, for any dimension $n\geq 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying $\mathrm{Sec}_g\leq\bar k\leq 0$ and $\mathrm{Ric}_g\geq (n-1)\underline k$, then its isoperimetric profile is at least that of the model space of curvature $\bar k$ for small enough volumes. We also reduce the proof of the generalized Cartan-Hadamard conjecture to establishing rigidity in the equality case.

4. Very close Same question Cited by this paper Not relevant

   Total Curvature and the Isoperimetric Inequality in Cartan–Hadamard Manifolds

   M. Ghomi, J. Spruck · 2019 · Journal of Geometric Analysis · cited 38 times · Not summarized yet

   The paper applies level-set total-curvature comparisons to isoperimetric problems in spaces of nonpositive curvature.

   Abstract

   We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.

5. Close Groundwork for this paper Cited by this paper Not relevant

   Alexandrov geometry: foundations

   Stephanie Alexander, V. Kapovitch, A. Petrunin · 2019 · cited 31 times · Not summarized yet

   This book develops the structure theory of Alexandrov spaces with curvature bounds, the setting for the paper's Alexandrov-space theorem.

   Abstract

   Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the b…

6. Close Broader context Not relevant

   No Connectedness Lemma for Alexandrov spaces

   Jan Nienhaus, Dennis Wulle · 2026 · Not summarized yet

   The note gives counterexamples to a connectedness lemma in Alexandrov spaces, a neighboring topic in Alexandrov geometry.

   Abstract

   Since its introduction in 2003, Wilking's Connectedness Lemma has become a powerful tool in the study of manifolds with positive sectional curvature. In recent work of the authors, this has been extended to orbifolds. The purpose of this note is to exhibit counterexamples in the class of Alexandrov spaces for the Connectedness Lemma in both its ordinary and dual formulation. This answers a 2007 question of Wilking.

7. Close Groundwork for this paper Cited by this paper Used in depth by this paper Not relevant

   Remarks on Manifolds with Two-Sided Curvature Bounds

   V. Kapovitch, A. Lytchak · 2021 · Analysis and Geometry in Metric Spaces · cited 22 times · Not summarized yet

   The paper discusses distance functions and positive-reach sets under two-sided curvature bounds, relevant to the paper's Alexandrov-space setting.

   Abstract

   Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

8. Close Broader context Not relevant

   A note on "The Cartan-Hadamard conjecture and the Little Prince"

   Spyridon Michalakis · 2017 · Differential Geometry · preprint, not peer reviewed · Not summarized yet

   The note supplies elementary proofs of lemmas used to derive isoperimetric inequalities in Cartan-Hadamard manifolds.

   Abstract

   We provide elementary proofs of Lemmas 7.1 and 7.4 appearing in "The Cartan-Hadamard conjecture and the Little Prince", by B. Kloeckner and G. Kuperberg. The Lemmas play an important role in the derivation of novel isoperimetric inequalities. The original proofs relied on Sage, a symbolic algebra package, to factor certain algebraic varieties into irreducible components.

Prior work it builds on

1. Close* Groundwork for this paper Used in depth by this paper Not relevant

   Metric Spaces of Non-Positive Curvature

   M. Bridson, A. Haefliger · 1999 · cited 5,980 times · Not summarized yet

   This book concerns metric spaces of nonpositive curvature, a broad geometric framework relevant to the paper's curvature assumptions.

2. Related Groundwork for this paper Used in depth by this paper Not relevant

   The sharp $L^2$-Michael--Simon inequality

   Jeffrey S. Case, D. Mengesha · 2026 · cited 1 time · Not summarized yet

   The paper establishes a sharp Michael-Simon inequality for immersed submanifolds, related to the paper's sharp hypersurface inequality.

   Abstract

   Let $n \geq 3$. We prove that if $j \colon (\Sigma^n,g) \to (\mathbb{R}^N,dx^2)$ is an isometric immersion, then \begin{equation*} \int_\Sigma \left( \lvert \nabla u \rvert^2 + \frac{n(n-2)}{4}\lvert H \rvert^2 u^2 \right) \geq \frac{n(n-2)}{4}\mathrm{Vol}(S^n)^{2/n}\left( \int_\Sigma \lvert u \rvert^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}} \end{equation*} for all $u \in W^{1,2}(\Sigma)$, and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael--Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive …

3. Close* Groundwork for this paper Not relevant

   Regularity of isoperimetric hypersurfaces in Riemannian manifolds

   F. Morgan · 2003 · Transactions of the American Mathematical Society · cited 177 times · Not summarized yet

   The work concerns regularity of isoperimetric hypersurfaces, which underlie the paper's analysis of isoperimetric regions.

4. Related Broader context Not relevant

   The isoperimetric inequality for a minimal submanifold in Euclidean space

   S. Brendle · 2019 · arXiv: Differential Geometry · preprint, not peer reviewed · cited 112 times · Not summarized yet

   The paper proves a sharp isoperimetric inequality for minimal submanifolds in Euclidean space rather than for regions in Cartan-Hadamard spaces.

   Abstract

   We prove an isoperimetric inequality which holds for minimal submanifolds in Euclidean space of arbitrary dimension and codimension. Our estimate is sharp if the codimension is at most $2$.

5. Related* Groundwork for this paper Not relevant

   Sobolev and isoperimetric inequalities for riemannian submanifolds

   D. Hoffman, J. Spruck · 2010 · Communications on Pure and Applied Mathematics · cited 322 times · Not summarized yet

   The work concerns Sobolev and isoperimetric inequalities on Riemannian submanifolds, related to the paper's hypersurface inequality.