Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form

Shanlin Guan; Yuhang Zhao

Home/Archive/arXiv

arXiv · 2610.18947

Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form

Shanlin Guan·Yuhang Zhao

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $X:M^3\to\R^4$ be a complete, connected self-shrinker without boundary, and suppose that the squared norm $S$ of its second fundamental form is constant. We prove that $S=0$ or $S=1$. Accordingly, the image is a hyperplane through the origin or a generalized cylinder $\mathbb S^k(\sqrt{k})\times\R^{3-k}$, $1\le k\le3$, and the immersion is a Riemannian covering onto its image.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Shanlin Guan, Yuhang Zhao. 2026-10-13. Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form. https://arxiv.org/abs/2610.18947

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

KEEP EXPLORING

Related papers

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↗

arXiv2026

Curvature of left-invariant complex Finsler metric on Lie groups

Let $G$ be an even-dimensional connected real Lie group admitting a left-invariant complex structure $J$. Let $\mathfrak g$ be the Lie algebra of $G$, and $\mathfrak g^{1,0}$ the $(1,0)$-part of the complexification $\mathfrak g^{\mathbb{C}}$ of $\mathfrak g$. We derive explicit formulas for the holomorphic sectional curvature and holomorphic bisectional curvature of any left-invariant strongly pseudoconvex complex Finsler metric $F$ on $G$, expressed in terms of the complex Lie algebra $\mathfrak g^{1,0}$. We also obtain a necessary and sufficient condition for $F$ to be a Kähler-Finsler metric (resp. weakly Kähler-Finsler metric). As an application, we prove a rigidity result: if $G$ is a complex Lie group and $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a Kähler-Berwald metric if and only if $G$ is an abelian complex Lie group.

math.DG↗