Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.30803

Einstein four-manifolds of positive sectional curvature and ADM mass

Matthew Gursky·Andrea Malchiodi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that closed Einstein four-manifolds $(M,g)$ with positive sectional curvature and Euler characteristic and signature satisfying $2χ(M)-3|τ(M)|\le4$ must be homothetically isometric to the round $S^4$ or to $\mathbb{CP}^2$ with the Fubini-Study metric. The proof relies on an upper bound on the ADM mass of conformal blow-ups of $(M,g)$, combined with a lower one from \cite{GM}.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Matthew Gursky, Andrea Malchiodi. 2026-09-25. Einstein four-manifolds of positive sectional curvature and ADM mass. https://arxiv.org/abs/2609.30803

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

KEEP EXPLORING

Related papers

arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗
arXiv2026

A quadratic lower bound for the number of minimal geodesics

A minimal geodesic on a Riemannian manifold is a geodesic defined on $\mathbb{R}$ that lifts to a globally distance minimizing curve on the universal covering. Bangert proved that there is a lower bound for the number of geometrically distinct minimal geodesics of closed Riemannian manifolds that is linear in the first Betti number, using the stable norm unit ball on the first homology. We refine this method to obtain a quadratic lower bound.

math.DG↗
arXiv2026

Morse index and nullity of rotationally symmetric $Y$-singular minimal surfaces

We compute the Morse index and nullity of a one-parameter family of rotationally symmetric minimal $Y$-singular surfaces in $\mathbb R^3$ whose singular set is a circle. The second variation is studied on a weighted energy space adapted to the noncompact faces and decomposed into angular Fourier modes. The resulting modewise problem separates the fixed-junction Dirichlet contribution from a finite-dimensional junction form. The exceptional resonant radial mode is treated directly. We prove that the $Y$-catenoid has index one and nullity three, while $Y_α$ has index two and nullity five for $0<α\leqπ/3$.

math.DG↗