Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36563

$O(3)$-symmetry of four-dimensional steady Ricci solitons

Hanbing Fang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that a complete connected four-dimensional gradient steady Ricci soliton with positive curvature operator that is $κ$-noncollapsed at all scales admits an effective isometric $O(3)$ action preserving the potential function and having principal orbits diffeomorphic to $S^2$.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hanbing Fang. 2026-10-27. $O(3)$-symmetry of four-dimensional steady Ricci solitons. https://arxiv.org/abs/2610.36563

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

KEEP EXPLORING

Related papers

arXiv2026

Examples of topological sphere $λ$-self-expander

A $λ$-self-expander $F: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, for sufficiently large $λ$, we construct an immersed non-embedded spherical $λ$-self-expander in $\mathbb{R}^{n+1}$ with the mean curvature $H>0$. Moreover, we obtain the explicit solution for $n=2$.

math.DG↗
arXiv2026

Characterizations of the plane and the catenoid as capillary minimal surfaces

In this paper, we prove that an immersed annular capillary minimal surface outside the unit ball in $\mathbb{R}^3$ with one complete embedded end and finite total curvature must be either part of the plane or part of the catenoid. The proofs are based on the use of the Weierstrass data through the Hopf differential and associated holomorphic functions. For a planar end, we control the second fundamental form, while for a catenoidal end, we analyze the Gauss map in more detail using the boundary condition. We also prove nonexistence results for annular free boundary minimal surfaces with one end asymptotic to the end of Enneper's surface, under appropriate conditions. Finally, we establish an exterior analogue of Choe's result on capillary minimal surfaces with vertices.

math.DG↗
arXiv2026

A characterization of Pfaffian embeddings from (2, 3, 5)- into flat (4, 7)-geometries

Given two smooth manifolds with tangent subbundle distributions, an embedding is Pfaffian if its differential sends the distribution on the source into the distribution on the target. In this paper, we consider the question of existence of Pfaffian embeddings in the specific case where the source is a (2,3,5)-manifold, the target is the 7-dimensional space of isotropic 2-planes in a 6-dimensional symplectic vector space, and the Pfaffian condition is that the derived 3-distribution on the source be mapped into the natural 4-distribution on the target. This is one of the simpler non-trivial cases of the general question on existence of Pfaffian embeddings, but already the answer here requires solution of an interesting differential equation. It turns out that a generic (2,3,5)-manifold does not embed, the first obstruction being the fact that a Pfaffian embeddable (2,3,5)-manifold necessarily has a double root for its Cartan quartic at each point. We determine a complete characterization of embeddable (2,3,5)-manifolds in terms of their associated Cartan geometries, which characterization depends on higher order (non-harmonic) curvature as well.

math.DG↗