PreprintCC BY-NC-ND 4.0arXiv
Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and S≤ 1
Abstract
Let X:Mn→ℝn+p be an n-dimensional complete self-shrinker in arbitrary codimension p≥ 1. Suppose that the scalar curvature R is a positive constant and that the squared norm S of the second fundamental form satisfies S≤ 1. We prove that S≡ 1 and that X is isometric to either the round sphere Sn(√n) or the standard generalized cylinder Sk(√k)×ℝn−k, 2≤ k≤ n−1. In particular, no new higher-codimension examples occur under these assumptions. The proof uses an algebraic lemma which yields a uniform positive lower bound for the Bakry--Émery Ricci curvature, together with the comparison theorem of Wei--Wylie, the equivalence of finite Gaussian volume and polynomial volume growth due to Cheng--Zhou, and the gap theorem of Cao--Li.
Related
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BibTeX
@misc{guo2026complete,
title = {{Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and $S\le 1$}},
author = {Shunzi Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19464},
eprint = {2609.19464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19464},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and S≤ 1
AU - Shunzi Guo
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.19464
UR - https://arxiv.org/abs/2609.19464
AB - Let X:M^n→ℝ^n+p be an n-dimensional complete self-shrinker in arbitrary codimension p≥ 1. Suppose that the scalar curvature R is a positive constant and that the squared norm S of the second fundamental form satisfies S≤ 1. We prove that S≡ 1 and that X is isometric to either the round sphere S^n(√n) or the standard generalized cylinder S^k(√k)×ℝ^n−k, 2≤ k≤ n−1. In particular, no new higher-codimension examples occur under these assumptions. The proof uses an algebraic lemma which yields a uniform positive lower bound for the Bakry--Émery Ricci curvature, together with the comparison theorem of Wei--Wylie, the equivalence of finite Gaussian volume and polynomial volume growth due to Cheng--Zhou, and the gap theorem of Cao--Li.
ER -