DOI: 10.1515/crelle-2026-0091 ISSN: 0075-4102

Estimates on scalar curvature of self-shrinkers

Qing-Ming Cheng, Fengjiang Li, Guoxin Wei
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Abstract

In this paper, we study ๐‘›-dimensional complete self-shrinkers in

R n + 1 \mathbb{R}^{n+1}
with constant squared norm ๐‘† of the second fundamental form. We partially resolve the conjecture on ๐‘›-dimensional complete self-shrinkers in
R n + 1 \mathbb{R}^{n+1}
with constant squared norm ๐‘† of the second fundamental form. Furthermore, if the scalar curvature of an ๐‘›-dimensional self-shrinker is constant, then we prove that the scalar curvature ๐‘… satisfies
R โ‰ค n โˆ’ 1 R\leq n-1
. We also classify ๐‘›-dimensional complete self-shrinkers in
R n + 1 \mathbb{R}^{n+1}
with non-negative constant scalar curvature.

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