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Journal articles on the topic 'Euclidean mean curvature operator'

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Author: Grafiati

Published: 11 March 2023

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1

Yang, Dan, Jinchao Yu, Jingjing Zhang, and Xiaoying Zhu. "A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $." AIMS Mathematics 7, no. 1 (2022): 39–53. http://dx.doi.org/10.3934/math.2022003.

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<abstract><p>A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.</p></abstract>

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2

Pashaie, Firooz. "On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$." Tamkang Journal of Mathematics 51, no. 4 (November 1, 2020): 313–32. http://dx.doi.org/10.5556/j.tkjm.51.2020.3188.

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A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

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3

Chen, Bang-Yen. "Mean curvature and shape operator of isometric immersions in real-space-forms." Glasgow Mathematical Journal 38, no. 1 (January 1996): 87–97. http://dx.doi.org/10.1017/s001708950003130x.

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According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

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4

Pashaie, Firooz. "Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1." Proyecciones (Antofagasta) 40, no. 3 (June 1, 2021): 711–19. http://dx.doi.org/10.22199/issn.0717-6279-3584.

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In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

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5

Güler, Erhan, Hasan Hacısalihoğlu, and Young Kim. "The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space." Symmetry 10, no. 9 (September 12, 2018): 398. http://dx.doi.org/10.3390/sym10090398.

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We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

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6

HEJUN, SUN, and QI XUERONG. "EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN." Glasgow Mathematical Journal 53, no. 2 (December 8, 2010): 321–32. http://dx.doi.org/10.1017/s0017089510000728.

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AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

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7

Lee, Jae Won, Dong-Soo Kim, Young Ho Kim, and Dae Won Yoon. "Generalized null 2-type immersions in Euclidean space." Advances in Geometry 18, no. 1 (January 26, 2018): 27–36. http://dx.doi.org/10.1515/advgeom-2017-0029.

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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.

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8

CHENG, QING-MING, and YEJUAN PENG. "ESTIMATES FOR EIGENVALUES OF $\mathfrak L$ OPERATOR ON SELF-SHRINKERS." Communications in Contemporary Mathematics 15, no. 06 (November 19, 2013): 1350011. http://dx.doi.org/10.1142/s0219199713500119.

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In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator [Formula: see text], which is introduced by Colding and Minicozzi in [Generic mean curvature flow I; generic singularities, Ann. Math.175 (2012) 755–833], on an n-dimensional compact self-shrinker in R n+p. Estimates for eigenvalues of the differential operator [Formula: see text] are obtained. Our estimates for eigenvalues of the differential operator [Formula: see text] are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator [Formula: see text] on a bounded domain with a piecewise smooth boundary in an n-dimensional complete self-shrinker in R n+p. For Euclidean space R n, the differential operator [Formula: see text] becomes the Ornstein–Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein–Uhlenbeck operator.

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9

Mohammadpouri, Akram, and Firooz Pashaei. "$L_r$-biharmonic hypersurfaces in $\mathbb{E}^4$." Boletim da Sociedade Paranaense de Matemática 38, no. 5 (March 31, 2019): 9–18. http://dx.doi.org/10.5269/bspm.v38i5.38484.

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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.

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10

Shen, Wenguo. "Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin." Journal of Function Spaces 2020 (April 13, 2020): 1–11. http://dx.doi.org/10.1155/2020/9801931.

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In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

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11

COLARES, ANTONIO GERVASIO, and FERNANDO ENRIQUE ECHAIZ-ESPINOZA. "CONSTANT SCALAR CURVATURE HYPERSURFACES WITH SECOND-ORDER UMBILICITY." Glasgow Mathematical Journal 51, no. 2 (May 2009): 219–41. http://dx.doi.org/10.1017/s0017089508004643.

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AbstractWe extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres $S^{k}(\frac{1}{\sqrt{2}})\times S^{k}(\frac{1}{\sqrt{2}})$ embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

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12

Fu, Yu, and Lan Li. "A Class of Weingarten Surfaces in Euclidean 3-Space." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/398158.

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The class of biconservative surfaces in Euclidean 3-space𝔼3are defined in (Caddeo et al., 2012) by the equationA(grad H)=-H grad Hfor the mean curvature functionHand the Weingarten operatorA. In this paper, we consider the more general case that surfaces in𝔼3satisfyingA(grad H)=kH grad Hfor some constantkare called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in𝔼3.

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13

Bereanu, Cristian, Petru Jebelean, and Jean Mawhin. "Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces." Mathematische Nachrichten 283, no. 3 (February 26, 2010): 379–91. http://dx.doi.org/10.1002/mana.200910083.

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14

Lucas, Pascual, and Hector Fabián Ramírez-Ospina. "Hypersurfaces in pseudo-Euclidean spaces satisfying a linear condition on the linearized operator of a higher order mean curvature." Differential Geometry and its Applications 31, no. 2 (April 2013): 175–89. http://dx.doi.org/10.1016/j.difgeo.2013.01.002.

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15

Baikoussis, Christos, and David E. Blair. "On the Gauss map of ruled surfaces." Glasgow Mathematical Journal 34, no. 3 (September 1992): 355–59. http://dx.doi.org/10.1017/s0017089500008946.

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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.

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16

Bereanu, C., P. Jebelean, and J. Mawhin. "Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces." Proceedings of the American Mathematical Society 137, no. 01 (July 1, 2008): 161–69. http://dx.doi.org/10.1090/s0002-9939-08-09612-3.

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17

Mohammadpouri, Akram. "Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map." Boletim da Sociedade Paranaense de Matemática 36, no. 3 (July 1, 2018): 207–17. http://dx.doi.org/10.5269/bspm.v36i3.31263.

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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.

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18

Uyar, Düldül. "A new method for finding the shape operator of a hypersurface in Euclidean 4-space." Filomat 32, no. 17 (2018): 5827–36. http://dx.doi.org/10.2298/fil1817827u.

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In this paper, by taking a Frenet curve lying on a parametric or implicit hypersurface and using the extended Darboux frame field of this curve, we give a new method for calculating the shape operator?s matrix of the hypersurface depending on the extended Darboux frame curvatures. This new method enables us to obtain the Gaussian and mean curvatures of the hypersurface depending on the geodesic torsions of the curve and the normal curvatures of the hypersurface.

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19

Mendonça, Bruno, and Ruy Tojeiro. "Umbilical Submanifolds of Sn × R." Canadian Journal of Mathematics 66, no. 2 (April 1, 2014): 400–428. http://dx.doi.org/10.4153/cjm-2013-003-3.

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AbstractWe give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of ×ℝ, extending the classification of umbilical surfaces in ×ℝ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in × ℝ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic ×ℝ or ×ℝ, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of × R and ℍn × ℝ. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor ℝ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in ×ℝ and ℍn×ℝ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of × ℝ and ℍn × ℝ.

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20

Ache, Antonio G., and Micah W. Warren. "Approximating coarse Ricci curvature on submanifolds of Euclidean space." Advances in Geometry 22, no. 2 (April 1, 2022): 215–43. http://dx.doi.org/10.1515/advgeom-2022-0002.

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Abstract For an embedded submanifold Σ ⊂ ℝ N , Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].

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21

Tunçer, Yılmaz, Dae Won Yoon, and Murat Kemal Karacan. "Weingarten and Linear Weingarten Type Tubular Surfaces in." Mathematical Problems in Engineering 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/191849.

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We study tubular surfaces in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature, and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Euclidean 3-space.

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22

Song, Chong, and Jun Sun. "Skew mean curvature flow." Communications in Contemporary Mathematics 21, no. 01 (January 28, 2019): 1750090. http://dx.doi.org/10.1142/s0219199717500900.

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The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space [Formula: see text]. A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest.

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23

Ulucan, Neslihan, and Mahmut Akyigit. "Offset Ruled Surface in Euclidean Space with Density." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 1 (March 1, 2021): 219–33. http://dx.doi.org/10.2478/auom-2021-0015.

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Abstract In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

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24

Mebarki, N., and O. Nemoul. "A curvature operator for a regular tetrahedron shape in LQG." International Journal of Geometric Methods in Modern Physics 16, no. 06 (June 2019): 1950095. http://dx.doi.org/10.1142/s0219887819500956.

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An alternative approach introducing a 3-dimensional (3D) Ricci scalar curvature quantum operator given in terms of volume and area as well as new edge length operators is proposed. An example of monochromatic 4-valent node intertwiner state (equilateral tetrahedra) is studied and the scalar curvature measure for a regular tetrahedron shape is constructed. It is shown that all regular tetrahedron states are in the negative scalar curvature regime and for the semi-classical limit the spectrum is very close to the Euclidean regime.

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25

Ghazal, Tahsin, and Sharief Deshmukh. "Submanifolds of Euclidean space with parallel mean curvature vector." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 533–36. http://dx.doi.org/10.1155/s0161171291000728.

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The object of the paper is to study some compact submanifolds in the Euclidean spaceRnwhose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist ann-dimensional compact simply connected totally real submanifold inR2nwhose mean curvature vector is parallel. Then we show that then-dimensional compact totally real submanifolds of constant curvature and parallel mean curvature inR2nare flat. Finally we show that compact Positively curved submanifolds inRnwith parallel mean curvature vector are homology spheres. The last result in particular for even dimensional submanifolds implies that their Euler poincaré characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric immersion with parallel mean curvature vector inRn, answers the problem of Chern and Hopf

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26

Koh, Sung-Eun. "Sphere theorem by means of the ratio of mean curvature functions." Glasgow Mathematical Journal 42, no. 1 (March 2000): 91–95. http://dx.doi.org/10.1017/s0017089500010119.

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It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.1991 Mathematics Subject Classification 53C40, 53C20.

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27

Chen, Bang-Yen, and Oscar J. Garay. "Constant mean curvature hypersurfaces with constantδ-invariant." International Journal of Mathematics and Mathematical Sciences 2003, no. 67 (2003): 4205–16. http://dx.doi.org/10.1155/s0161171203304260.

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28

Ganchev, Georgi, and Velichka Milousheva. "Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space." Filomat 33, no. 4 (2019): 1135–45. http://dx.doi.org/10.2298/fil1904135g.

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We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

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29

Kapouleas, Nicolaos. "Constant mean curvature surfaces in Euclidean three-space." Bulletin of the American Mathematical Society 17, no. 2 (October 1, 1987): 318–21. http://dx.doi.org/10.1090/s0273-0979-1987-15575-3.

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30

Li, Guanghan, and Isabel Salavessa. "Forced convex mean curvature flow in Euclidean spaces." manuscripta mathematica 126, no. 3 (March 21, 2008): 333–51. http://dx.doi.org/10.1007/s00229-008-0181-z.

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31

Chen, Li, Xi Guo, and Qiang Tu. "Nonhomogeneous inverse mean curvature flow in Euclidean space." Proceedings of the American Mathematical Society 148, no. 10 (July 20, 2020): 4557–71. http://dx.doi.org/10.1090/proc/15099.

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32

Huang, Rongli. "Lagrangian mean curvature flow in pseudo-Euclidean space." Chinese Annals of Mathematics, Series B 32, no. 2 (January 25, 2011): 187–200. http://dx.doi.org/10.1007/s11401-011-0639-2.

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33

Barros, Manuel, and Oscar J. Garay. "Euclidean submanifolds with Jacobi mean curvature vector field." Journal of Geometry 58, no. 1-2 (March 1997): 15–25. http://dx.doi.org/10.1007/bf01222923.

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34

CASTRO, ILDEFONSO, and ANA M. LERMA. "TRANSLATING SOLITONS FOR LAGRANGIAN MEAN CURVATURE FLOW IN COMPLEX EUCLIDEAN PLANE." International Journal of Mathematics 23, no. 10 (October 2012): 1250101. http://dx.doi.org/10.1142/s0129167x12501017.

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Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones [Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom.84 (2010) 127–161] in dimension two. The simplest (non-trivial) example in our family is characterized as the only (non-totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.

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35

Burns, John M., and Michael J. Clancy. "SMYTH SURFACES AND THE DREHRISS." Proceedings of the Edinburgh Mathematical Society 48, no. 3 (September 15, 2005): 549–55. http://dx.doi.org/10.1017/s0013091504000598.

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AbstractIsometric deformations of immersed surfaces in Euclidean 3-space are studied by means of the drehriss. When the immersion is of constant mean curvature and the deformation preserves the mean curvature, we determine the drehriss explicitly in terms of the immersion and its Gauss map. These methods are applied to obtain an alternative classification of the Smyth surfaces, i.e. constant mean curvature immersions of the plane into Euclidean 3-space which admit the action of $S^1$ as a non-trivial group of internal isometries.

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36

Arslan, Kadri, Bengü Bayram, Betül Bulca, and Günay Öztürk. "Rotational submanifolds in Euclidean spaces." International Journal of Geometric Methods in Modern Physics 16, no. 02 (February 2019): 1950029. http://dx.doi.org/10.1142/s0219887819500294.

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The rotational embedded submanifold was first studied by Kuiper as a submanifold in [Formula: see text]. The generalized Beltrami submanifolds and toroidal submanifold are the special examples of these kind of submanifolds. In this paper, we consider [Formula: see text]-dimensional rotational embedded submanifolds in Euclidean [Formula: see text]-space [Formula: see text]. We give some basic curvature properties of this type of submanifolds. Further, we obtain some results related with the scalar curvature and mean curvature of these submanifolds. As an application, we give an example of rotational submanifold in [Formula: see text].

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37

Sawicz, Katarzyna. "Curvature properties of some class of hypersurfaces in Euclidean spaces." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 165–77. http://dx.doi.org/10.2298/pim141025008s.

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We determine curvature properties of pseudosymmetry type of hypersurfaces in Euclidean spaces En+1, n ? 5, having three distinct nonzero principal curvatures ?1, ?2 and ?3 of multiplicity 1, p and n-p-1, respectively. For some hypersurfaces having this property the sum of ?1, ?2 and ?3 is equal to the trace of the shape operator of M. We present an example of such hypersurface.

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Sato, Naoki, and Michio Yamada. "Vorticity equation on surfaces with arbitrary topology embedded in three-dimensional Euclidean space." Journal of Mathematical Physics 63, no. 9 (September 1, 2022): 093101. http://dx.doi.org/10.1063/5.0080453.

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We derive the vorticity equation for an incompressible fluid on a two-dimensional surface with an arbitrary topology, embedded in three-dimensional Euclidean space and arising from a first integral of the flow, by using a tailored Clebsch parameterization of the velocity field. In the inviscid limit, we identify conserved surface energy and enstrophy and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for Navier–Stokes equations on a Riemannian manifold and calculate the diffusion operator by requiring that flows corresponding to Killing fields of the Riemannian metric are not subject to dissipation. For an embedded surface, this leads to a diffusion operator, including derivatives of the stream function across the surface. In the second case, using an analogy with the Poisson equation for the Newtonian gravitational potential in general relativity, we construct a diffusion operator taking into account the Ricci scalar curvature of the surface. The resulting vorticity equation is two-dimensional, and the corresponding diffusive equilibria minimize dissipation under the constraint of curvature energy.

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39

ARSLAN, KADRI, ALIM SUTVEREN, and BETUL BULCA. "Rotational λ – hypersurfaces in Euclidean spaces." Creative Mathematics and Informatics 30, no. 1 (February 15, 2021): 29–40. http://dx.doi.org/10.37193/cmi.2021.01.04.

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Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtained some results related with rotational hypersurfaces in Euclidean 4-space \mathbb{R}^{4} to become self-shrinkers. Furthermore, we classify the general rotational \lambda -hypersurfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational \lambda -hypersurfaces in \mathbb{R}^{4}.

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40

Kapouleas, Nicolaos. "Complete Constant Mean Curvature Surfaces in Euclidean Three-Space." Annals of Mathematics 131, no. 2 (March 1990): 239. http://dx.doi.org/10.2307/1971494.

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Wang, Baofu, and An-Min Li. "Euclidean Complete Hypersurfaces with Negative Constant Affine Mean Curvature." Results in Mathematics 52, no. 3-4 (September 2008): 383–98. http://dx.doi.org/10.1007/s00025-008-0320-6.

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42

Kapouleas, Nicolaos. "Compact constant mean curvature surfaces in Euclidean three-space." Journal of Differential Geometry 33, no. 3 (1991): 683–715. http://dx.doi.org/10.4310/jdg/1214446560.

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43

Li, Haozhao. "The volume-preserving mean curvature flow in Euclidean space." Pacific Journal of Mathematics 243, no. 2 (December 1, 2009): 331–55. http://dx.doi.org/10.2140/pjm.2009.243.331.

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Alías, Luis J., and Paolo Piccione. "Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres." Journal of Geometric Analysis 23, no. 2 (September 23, 2011): 677–708. http://dx.doi.org/10.1007/s12220-011-9260-6.

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45

Breiner, Christine, and Nikolaos Kapouleas. "Embedded constant mean curvature surfaces in Euclidean three-space." Mathematische Annalen 360, no. 3-4 (June 25, 2014): 1041–108. http://dx.doi.org/10.1007/s00208-014-1056-0.

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46

ABDELHEDI, WAEL, and HICHEM CHTIOUI. "PRESCRIBING MEAN CURVATURE ON 𝔹n." International Journal of Mathematics 21, no. 09 (September 2010): 1157–87. http://dx.doi.org/10.1142/s0129167x10006434.

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In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

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BAŞ, Selçuk, and Talat KÖRPINAR. "Modified Roller Coaster Surface in Space." Mathematics 7, no. 2 (February 19, 2019): 195. http://dx.doi.org/10.3390/math7020195.

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In this paper, a new modified roller coaster surface according to a modified orthogonal frame is investigated in Euclidean 3-space. In this method, a new modified roller coaster surface is modeled. Both the Gaussian curvature and mean curvature of roller coaster surfaces are investigated. Subsequently, we obtain several characterizations in Euclidean 3-space.

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Minlend, Ignace Aristide, Alassane Niang, and El hadji Abdoulaye Thiam. "Multiply-periodic hypersurfaces with constant nonlocal mean curvature." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 10. http://dx.doi.org/10.1051/cocv/2019047.

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We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.

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49

Moriya, Katsuhiro. "Darboux Transforms of a Harmonic Inverse Mean Curvature Surface." Geometry 2013 (April 7, 2013): 1–9. http://dx.doi.org/10.1155/2013/902092.

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The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäcklund transform. For a given isothermic harmonic inverse mean curvature surface, its classical Darboux transform is a harmonic inverse mean curvature surface. Then a transform of a solution to the Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution.

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Chen, Bang-Yen. "Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector." Mathematics 7, no. 8 (August 6, 2019): 710. http://dx.doi.org/10.3390/math7080710.

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The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

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