On complete submanifolds with parallel normalized mean curvature in product spaces

Santos, Fábio R. dos; Silva, Sylvia F. da

doi:10.1017/prm.2021.6

Cited by 3 publications

(2 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We end this section by recalling the following two results, which we shall use later in this paper. The first one is a Simons-type formula proved in [7,8]. It should be noticed that, for the sake of simplicity, in Proposition 1 and, in general, along this manuscript, we will naturally identify, at convenience, the Weingarten operator with its associated symmetric matrix.…”

Section: Preliminariesmentioning

confidence: 91%

Total mean curvature surfaces in the product space and applications

Albujer¹,

Silva²,

Santos³

2023

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $\mathcal{H}$ -surfaces in $\mathbb{S}^n\times\mathbb{R}$ , characterizing the cases where the equality is attained.

show abstract

Section: Preliminariesmentioning

confidence: 91%

Total mean curvature surfaces in the product space and applications

Albujer¹,

Silva²,

Santos³

2023

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Later on, the authors of [26] considered a compact minimal submanifold M n in S n 1 (c) × R n 2 and obtained several rigidity results depending on the Ricci curvature, the squared length and the squared maximum norm of the second fundamental form on M n . Some other types of pinching theorems in general product spaces seem to be of interest and worth further discussion (refer to [27][28][29][30]).…”

Section: Introductionmentioning

confidence: 99%

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Qiu,

Zhan

2023

Mathematics

View full text Add to dashboard Cite

Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

show abstract

Integral inequalities for closed linear Weingarten submanifolds in the product spaces

SANTOS,

SILVA,

SOUSA

2023

An. Acad. Bras. Ciênc.

Self Cite

View full text Add to dashboard Cite

An integral inequality for closed linear Weingarten 𝑚-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces 𝑀 𝑛 (𝑐) × ℝ, 𝑛 > 𝑚 ≥ 4, where 𝑀 𝑛 (𝑐) is a space form of constant sectional curvature 𝑐 ∈ {−1, 1}, is proved. As an application is shown that the sharpness in this inequality is attained in the totally umbilical hypersurfaces, and in a certain family of standard product of the form 𝕊 1 (√1 − 𝑟 2 ) × 𝕊 𝑚−1 (𝑟) with 0 < 𝑟 < 1 when 𝑐 = 1. In the case where 𝑐 = −1, is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces. When 𝑚 = 2 and 𝑚 = 3, an integral inequality is also obtained with equality happening in the totally umbilical hypersurfaces.

show abstract

On complete submanifolds with parallel normalized mean curvature in product spaces

Cited by 3 publications

References 30 publications

Total mean curvature surfaces in the product space and applications

Total mean curvature surfaces in the product space and applications

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Integral inequalities for closed linear Weingarten submanifolds in the product spaces

Contact Info

Product

Resources

About