Complete 3-dimensional λ-translators in the Euclidean space ℝ4

2021 ◽  
pp. 1-54
Author(s):  
Zhi Li ◽  
Guoxin Wei ◽  
Gangyi Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
3 Dimensional ◽  
The Second Fundamental Form

In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

A note on rigidity of spacelike self-shrinkers

2020 ◽  
Vol 31 (05) ◽  
pp. 2050035
Author(s):  
Yong Luo ◽  
Hongbing Qiu
Keyword(s):  
Euclidean Space ◽  
Growth Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Integral Method ◽  
Acta Math ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
A Minor

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82].

scholarly journals The length of the second fundamental form, a tangency principle and applications

2004 ◽  
Vol 76 (1) ◽  
pp. 1-7
Author(s):  
Francisco X. Fontenele ◽  
Sérgio L. Silva
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Positive Definite ◽  
Ambient Space ◽  
Basic Tool ◽  
The Second Fundamental Form ◽  
Tangency Principle

In this paper we prove a tangency principle (see Fontenele and Silva 2001) related with the length of the second fundamental form, for hypersurfaces of an arbitrary ambient space. As geometric applications, we make radius estimates of the balls that lie in some component of the complementary of a complete hypersurface into Euclidean space, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The basic tool established here is that some operator is elliptic at points where the second fundamental form is positive definite.

scholarly journals Some relations between differential geometric invariants and topological invariants of submanifolds

1976 ◽  
Vol 60 ◽  
pp. 1-6 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Vector Fields ◽  
Tangent Vector ◽  
Second Fundamental Form ◽  
Dimensional Euclidean Space ◽  
Topological Invariants ◽  
Dimensional Manifold ◽  
Geometric Invariants ◽  
The Second Fundamental Form

Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by(1.1) ∇̃XY = ∇XY + h(X,Y).

Immersions with Semi-Definite Second Fundamental Forms

1975 ◽  
Vol 27 (3) ◽  
pp. 610-617 ◽  
Author(s):  
Leo B. Jonker
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Form ◽  
Isometric Immersion ◽  
Tangent Space ◽  
Normal Component ◽  
Second Fundamental Form ◽  
Normal Space ◽  
The Second Fundamental Form ◽  
Image Position

Let M be a. complete connected Riemannian manifold of dimension n and let £:M → Rn+k be an isometric immersion into the Euclidean space Rn+k. Let ∇ be the connection on Mn and let be the Euclidean connection on Rn+k. Also letdenote the second fundamental form B(X, Y) = (xY)→. Here TP(M) denotes the tangent space at p, NP(M) the normal space and (…)→ the normal component.

scholarly journals Characterizations of the sphere by the mean II-curvature

1981 ◽  
Vol 23 (2) ◽  
pp. 249-253 ◽  
Author(s):  
George Stamou
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gaussian Curvature ◽  
Convex Surface ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Second Fundamental Form ◽  
The Mean

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

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