Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.12352

A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds

Aghil Alaee·Marcus Khuri·Hari Kunduri
View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish sharp lower bounds for the mass of asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) toric 4-manifolds, in terms of equilibrium geometries consisting of gravitational instantons. More precisely, the mass of a complete ALE or ALF toric 4-manifold with nonnegative scalar curvature is bounded below by a sum comprised of the following quantities: the mass of the corresponding toric gravitational instanton having the same orbit space (rod) structure as the original ALE/ALF manifold, and an expression determined by the conical angle defects of totally geodesic 2-spheres within the instanton that serve as generators for its second homology. The inequality may be generalized to the situation in which the ALE/ALF manifold also possesses conical singularities as well as orbifold singularities, and it suggests a refined notion of `total mass' in which the result simply states that the total mass of the ALE/ALF manifold is not less than that of the corresponding gravitational instanton. Furthermore, we prove rigidity for these statements, namely the inequality is saturated only when the ALE/ALF manifold is Ricci flat and in fact agrees with the corresponding instanton. These results may be viewed in the context of positive mass theorems, providing an explanation of how positivity can fail in the ALE/ALF setting. Moreover, the main theorem may be interpreted as yielding a variational characterization of the relevant toric gravitational instantons.

Explore related subjects

math.DGgr-qc

Keep this discovery

BibTeXRIS

Aghil Alaee, Marcus Khuri, Hari Kunduri. 2026-05-12. A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds. https://arxiv.org/abs/2605.12352

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

KEEP EXPLORING

Related papers

arXiv2026

The $q$-deformed cross-ratio: modular invariants and Coxeter friezes

We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

math.DG↗
arXiv2026

The Cartan-Hadamard conjecture in dimension five

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 new proof in dimension $3$.

math.DG↗
arXiv2026

On static manifolds with boundary admitting a nowhere-vanishing static potential

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

math.DG↗