Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.29842

Rigidity theorems of complete self-shrinkers in the Euclidean space

Zhi Li·Guoxin Wei·Cheng Xing
View original source ↗Read source document ↗Export metadata ↓

Abstract

We classify complete Lagrangian self-shrinking surfaces in $\mathbb{R}^{4}$ with constant squared norm of the mean curvature vector, without assuming an upper bound for the second fundamental form. Their images are a plane through the origin, the round cylinder $S^1(1)\times \mathbb{R}$, or the Clifford torus $S^1(1)\times S^1(1)$. More generally, a complete self-shrinking surface in $\mathbb{R}^{4}$ with constant $|\vec{H}|^2$ is either a plane, a round cylinder, or a complete minimal immersion into $S^3(\sqrt2)$. The proof uses the function $|x|^2-|\vec{H}|^2$, the Bochner formula, and completeness.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhi Li, Guoxin Wei, Cheng Xing. 2026-10-21. Rigidity theorems of complete self-shrinkers in the Euclidean space. https://arxiv.org/abs/2610.29842

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

KEEP EXPLORING

Related papers

arXiv2026

On electrostatic manifolds with boundary

Static manifolds with boundary were recently introduced in the context of the problem of prescribing both the scalar curvature of a Riemannian manifold and the mean curvature of its boundary. These manifolds are also of interest from the perspective of general relativity. In this article, we introduce electrostatic manifolds with boundary as a natural generalization of static manifolds with boundary in the presence of a nonzero electric field. We also introduce Robin electrostatic quadruples, extending the notion of Robin static triples previously introduced by the second author. We establish several uniqueness theorems for suitable regions of the Reissner-Nordström manifold. Finally, we characterize the criticality of the first eigenvalue of an associated mixed Robin-Dirichlet Schrödinger operator under admissible volume-preserving variations in terms of the equality of the surface gravities along the Dirichlet boundary components.

math.DG↗
arXiv2026

Non-existence of cohomogeneity one Einstein metrics of two summands

We prove the non-existence of cohomogeneity one Einstein metrics on a class of compact manifolds arising as double disk bundles, whose principal orbits split into two inequivalent irreducible summands. The proof uses a phase space barrier argument that yields a new obstruction to the existence of closed cohomogeneity one Einstein metrics, even when the principal orbits admit homogeneous Einstein metrics.

math.DG↗
arXiv2026

$SU(2)$ Yang-Mills-Higgs functional with Higgs self-interaction on $3$-manifolds

Given a constant $λ>0$, for any parameter $\varepsilon>0$ we study critical points of the Yang--Mills--Higgs energy \[ \mathcal{Y}_{\varepsilon}(\nabla,Φ) = \int_M \left(\varepsilon^2|F_{\nabla}|^2 + |\nablaΦ|^2 + \fracλ{4\varepsilon^2}(1-|Φ|^2)^2\right)\mathrm{vol}_g, \] defined for pairs $(\nabla,Φ)$, where $\nabla$ is a connection on an $SU(2)$-bundle over an oriented Riemannian $3$-manifold $(M^3, g)$, and $Φ$ a section of the associated adjoint bundle. When $M$ is closed, we use a $3$-parameter min-max construction to produce, for $\varepsilon\ll_M 1$, non-trivial critical points in the energy regime \[ 1 \lesssim_λ\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon},Φ_{\varepsilon}) \lesssim_{λ, M} 1. \] When $b_1(M)=0$, these critical points are irreducible: $\nabla_{\varepsilon}Φ_{\varepsilon}\neq 0$. Next, assuming $M$ has bounded geometry (not necessarily compact), and given critical points with $\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon}, Φ_{\varepsilon})$ uniformly bounded, we show that as $\varepsilon\to 0$, the energy measures $\varepsilon^{-1}e_{\varepsilon}(\nabla_{\varepsilon}, Φ_{\varepsilon}) \mathrm{vol}_{g}$ converge subsequentially to \[ |h|^2 \mathrm{vol}_g + \sum_{x \in S}Θ(x)δ_{x}, \] where $h$ is an $L^2$ harmonic $1$-form, $S$ a finite set and each $Θ(x)$ equals the energy of a finite collection of $\mathcal{Y}_{1}$-critical points on $\mathbb{R}^3$. Finally, the estimates involved also lead to an energy gap for critical points on $3$-manifolds with bounded geometry. As a byproduct of our results, we deduce the existence of non-trivial $\mathcal{Y}_{1}$-critical points over $\mathbb{R}^3$ for any $λ>0$.

math.DG↗