scholarly journals Hypersurfaces with Generalized 1-Type Gauss Maps

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 130 ◽  
Author(s):  
Dae Yoon ◽  
Dong-Soo Kim ◽  
Young Kim ◽  
Jae Lee
Keyword(s):  
Euclidean Space ◽  
Laplace Operator ◽  
Cylindrical Surface ◽  
Gauss Map ◽  
Dimensional Euclidean Space ◽  
Constant Vector ◽  
Gauss Maps ◽  
Tangent Developable ◽  
Conical Surfaces ◽  
The Laplace Operator

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.

Generalized null 2-type immersions in Euclidean space

2018 ◽  
Vol 18 (1) ◽  
pp. 27-36
Author(s):  
Jae Won Lee ◽  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Dae Won Yoon
Keyword(s):  
Euclidean Space ◽  
Curvature Vector ◽  
Dimensional Euclidean Space ◽  
Constant Vector ◽  
Smooth Functions ◽  
Developable Surfaces ◽  
Tangent Developable ◽  
The Mean ◽  
Conical Surfaces ◽  
The Laplace Operator

AbstractWe define generalized null 2-type submanifolds in them-dimensional Euclidean space 𝔼m. Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the conditionΔ H=f H+gCfor some smooth functionsf,gand a constant vectorCin 𝔼m, whereΔandHdenote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼3and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

scholarly journals On the Gauss map of ruled surfaces

1992 ◽  
Vol 34 (3) ◽  
pp. 355-359 ◽  
Author(s):  
Christos Baikoussis ◽  
David E. Blair
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Connected Surface ◽  
Image Position ◽  
The Laplace Operator ◽  
Special Case

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only ifwhere δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given byi.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

scholarly journals Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

2018 ◽  
Vol 36 (3) ◽  
pp. 207-217
Author(s):  
Akram Mohammadpouri
Keyword(s):  
Euclidean Space ◽  
Smooth Function ◽  
Mean Curvature ◽  
Gauss Map ◽  
Constant Vector ◽  
Newton Transformation ◽  
Linearized Operator

In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

scholarly journals The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 186
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Characteristic Polynomial ◽  
Fundamental Form ◽  
Shape Operator ◽  
Gauss Map ◽  
Dimensional Euclidean Space ◽  
Operator Matrix ◽  
Geometric Objects

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects.

scholarly journals The Principal Curvatures and the Third Fundamental Form of Dini-Type Helicoidal Hypersurface in 4-Space

2020 ◽  
pp. 62-68
Author(s):  
Erhan G¨uler
Keyword(s):  
Euclidean Space ◽  
Characteristic Polynomial ◽  
Fundamental Form ◽  
Shape Operator ◽  
Gauss Map ◽  
Dimensional Euclidean Space ◽  
Operator Matrix ◽  
Principal Curvatures ◽  
The Third ◽  
Third Fundamental Form

We consider the principal curvatures and the third fundamental form of Dini-type helicoidal hypersurface D(u, v, w) in the four dimensional Euclidean space E4. We find the Gauss map e of helicoidal hypersurface in E4. We obtain characteristic polynomial of shape operator matrix S. Then, we compute principal curvatures ki=1;2;3, and the third fundamental form matrix III of D.

scholarly journals The asymptotic distribution of eigenvalues of the Laplace operator in an unbounded domain

1976 ◽  
Vol 60 ◽  
pp. 7-33 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Euclidean Space ◽  
Boundary Conditions ◽  
Asymptotic Distribution ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Unbounded Domain ◽  
Distribution Of Eigenvalues ◽  
The Laplace Operator

This paper is devoted to the study of the asymptotic distribution of eigenvalues of the Laplace operator with zero boundary conditions in a quasi-bounded domain contained in Euclidean space R2.

scholarly journals $L_r$-biharmonic hypersurfaces in $\mathbb{E}^4$

2019 ◽  
Vol 38 (5) ◽  
pp. 9-18
Author(s):  
Akram Mohammadpouri ◽  
Firooz Pashaei
Keyword(s):  
Euclidean Space ◽  
Laplace Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
The Subject ◽  
The Laplace Operator

A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is said to be biharmonic if $\Delta^2x=0$, where $\Delta$ is the Laplace operator of $M^n$. Based on a well-known conjecture of Bang-Yen Chen, the only biharmonic hypersurfaces in $E^{n+1}$ are the minimal ones. In this paper, we study an extension of biharmonic hypersurfaces in 4-dimentional Euclidean space $\mathbb{E}^4$. A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is called $L_r$-biharmonic if $L_r^2x=0$, where $L_r$ is the linearized opereator of $(r + 1)$th mean curvature of $M^n$. Since $L_0=\Delta$, the subject of $L_r$-biharmonic hypersurface is an extension of biharmonic ones. We prove that any $L_2$-biharmonic hypersurface in $\mathbb{E}^4$ with constant $2$-th mean curvature is $2$-minimal. We also prove that any $L_1$-biharmonic hypersurfaces in $\mathbb{E}^4$ with constant mean curvature is $1$-minimal.

Oscillation Criteria for a Class of Perturbed Schrödinger Equations

1982 ◽  
Vol 25 (1) ◽  
pp. 71-77 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito
Keyword(s):  
Elliptic Equation ◽  
Euclidean Space ◽  
Exterior Domain ◽  
Continuous Functions ◽  
Oscillatory Behavior ◽  
Dimensional Euclidean Space ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
Image Position ◽  
The Laplace Operator

We are concerned with the oscillatory behavior of the second order elliptic equation1where Δ is the Laplace operator inn-dimensional Euclidean spaceRn,Eis an exterior domain inRn, andc:E × R → Randf:E → Rare continuous functions.A functionv : E − Ris called oscillatory inEifv(x) has arbitrarily large zeros, that is, the set {x∈E:v(x) = 0} is unbounded. For brevity, we say that equation (1) is oscillatory inEif every solutionu∈C2(E) of (1) is oscillatory inE.

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃 m

2021 ◽  
pp. 1-20
Author(s):  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Laplace Operator ◽  
Minkowski Spacetime ◽  
Upper Bounds ◽  
First Eigenvalue ◽  
Original Result ◽  
The First Eigenvalue ◽  
Spacelike Submanifold ◽  
Compact Submanifold ◽  
The Laplace Operator

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

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