Abhitosh Upadhyay, Nurettin Cenk Turgay
In this paper, we study biconservative hypersurfaces of index 2 in $\mathbb E^{5}_{2}$. We give the complete classification of biconservative hypersurfaces with diagonalizable shape operator at exactly three distinct principal curvatures. We also give an explicit example of biconservative hypersurfaces with four distinct principal curvatures.
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In this paper, we study biconservative hypersurfaces in $\mathbb S^{n}$ and $\mathbb H^{n}$. Further, we obtain complete explicit classification of biconservative hypersurfaces in $4$-dimensional Riemannian space form with exactly three distinct principal curvatures.
In this paper, we study biconservative hypersurfaces in the four dimensional Minkowski space $\mathbb E^4_1$. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in $\mathbb E^4_1$.
Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces $M^n$ with constant scalar curvature in a space form $N^{n+1}(c)$. We prove that every biconservative hypersurface with constant scalar curvature in $N^4(c)$ has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in $N^5(c)$ is ether an open part of a certain rotational hypersurface …
In this paper, we obtain some properties of biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with at most five distinct principal curvatures has constant mean curvature. Also, we investigate such type of hypersurface with constant length of second fundamental form having six distinct principal curvatures.
Compact spacelike biconservative hypersurfaces in de Sitter space
本文研究de Sitter空间中常数量曲率紧致类空双守恒超曲面在几何约束下的刚性性质，拓展了伪黎曼背景下该类超曲面的理解。
We present some general properties of biharmonic and biconservative submanifolds and then survey recent results on such hypersurfaces in space forms. We also propose an alternative version for a well-known result of Nomizu and Smyth for hypersurfaces by replacing the CMC hypothesis with the more general condition of biconservativity.
Rigidity results for compact biconservative hypersurfaces in space forms
本文研究了空间形式中紧致双调和超曲面的刚性问题，提供了两个已知刚性结果的替代证明，并通过将非负截面曲率假设替换为形状算子平方范数的估计，证明了新的刚性结果。
Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.
Biconservative hypersurfaces in space forms $\overline{M}^{\lowercase{n+1}}(\lowercase{c})$
针对空间形式中具四个不同主曲率、第二基本形式范数为常数的双保守超曲面，证明其均具有常平均曲率和常数量曲率。
A biconservative submanifold of a Riemannian manifold is a sub- manifold with divergence free stress-energy tensor with respect to bienergy. These are generalizations of biharamonic submanifolds. In 2013, B. Y. Chen and M.I. Munteanu proved that $δ(2)$-ideal and $δ(3)$-ideal biharmonic hypersurfaces in Euclidean space are minimal. In this paper, we generalize this result for $δ(2)$-ideal and $δ(3)$-ideal bisonservative hypersurfaces in Euclidean space. Also, we study $δ(4)$-ideal biconservative hypersurfaces in Euclidean space $\mathbb{E}^{6}$ having constant scalar curvature. We prove that su…
A classification of Biconservative Hypersurfaces in the Minkowski spaces
本文研究了平均曲率梯度为类光的Lorentzian双保守超曲面，通过严格分析Codazzi和Gauss方程，证明了此类超曲面在Minkowski空间中不存在。
Biconservative hypersurfaces are hypersurfaces with conservative stress-energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3-dimensional Bianchi-Cartan-Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)-invariant.
Closed Biconservative Hypersurfaces in Spheres
本文将空间形式中非 CMC 双保守旋转超曲面的轮廓曲线刻画为依赖维数的 $p$-弹性曲线，并通过闭合条件证明球面中存在离散双参数族的非 CMC 闭双保守超曲面，且它们均不可嵌入。
Compact Biconservative Hypersurfaces in Space Forms: Rigidity Without Scalar Curvature Assumptions
本文研究空间形式中紧致双守恒超曲面的刚性，通过引入新的无散度张量，在不需要常标量曲率或截面曲率假设的条件下建立了刚性结果。
Biconservative hypersurfaces in Euclidean Spaces
本文研究欧氏空间中的双守恒超曲面，先描述其定义，再构造这类仅双调和方程切向部分消失的等距浸入。
In this paper, we study biconservative surfaces with parallel normalized mean curvature vector in $\mathbb{E}^4$. We obtain complete local classification in $\mathbb{E}^4$ for a biconservative PNMCV surface. We also give an example to show the existence of PNMCV biconservative surfaces in $\mathbb{E}^4$.
Biconservative and Biharmonic Hypersurfaces in Product Spaces
本文研究乘积空间中的双守恒与双调和超曲面，给出等参与常角情形的完整描述，并证明分类、刚性及全局不存在性定理。
In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.
In this paper, we study biconservative surfaces with parallel normalized mean curvature vector field ($PNMC$) in the $4$-dimensional unit Euclidean sphere $\mathbb{S}^4$. First, we study the existence and uniqueness of such surfaces. We obtain that there exists a $2$-parameter family of non-isometric abstract surfaces that admit a (unique) $PNMC$ biconservative immersion in $\mathbb{S}^4$. Then, we obtain the local parametrization of these surfaces in the $5$-dimensional Euclidean space $\mathbb{E}^5$.
In this paper we consider the complete biconservative surfaces in Euclidean space $\mathbb{R}^3$ and in the unit Euclidean sphere $\mathbb{S}^3$. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the gradient of their mean curvature function is an eigenvector of the shape operator, and we are interested in studying local and global properties of such surfaces with non-constant mean curvature function. We determine the simply connected, complete Riemannian surfaces that admit biconservative immersions in $\mathbb{R}^3$ and $\mathbb{S}^3$. Moreover, such imm…
We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in 3-dimensional space forms.
Recent development in biconservative submanifolds
本文综述了双保守子流形在过去十年中的最新发展，涵盖了其从双调和子流形衍生而来的定义、与H-子流形的等价关系及相关研究成果。
We study the uniqueness of complete biconservative surfaces in the Euclidean space $\mathbb{R}^3$, and prove that the only complete biconservative regular surfaces in $\mathbb{R}^3$ are either $CMC$ or certain surfaces of revolution. In particular, any compact biconservative regular surface in $\mathbb{R}^3$ is a round sphere.
Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space
本文研究了四维双曲空间中非常数平均曲率（非CMC）且具有平行归一化平均曲率向量场（PNMC）的双保守曲面，通过超曲面模型提供了这些曲面的局部外在描述，并将其分为三种情况，从而完善了四维空间形式中非CMC、PNMC曲面的分类。
Biharmonic Hypersurfaces in Euclidean Spaces
本文研究欧氏空间中双调和超曲面，构造了主曲率方向可积的双守恒超曲面，并证明完整双调和超曲面当且仅当为极小超曲面。
In the present paper, we revisit the rigidity of hypersurfaces in Euclidean space. We highlight Darboux equation and give new proof of rigidity of hypersurfaces by energy method and maximal principle.
In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3Hrho = Krho^3-2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclide…
Biharmonic $δ(\lowercase{r})$-ideal hypersurfaces in Euclidean spaces are minimal
本文研究欧氏空间双调和子流形的陈邦彦猜想，证明n≥3时欧氏空间中δ(r)-理想双调和超曲面必为极小超曲面，推广了已有结果，还得到该条件下双保守超曲面为常平均曲率的结论。
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures. We also give explicit examples.
We study hypersurfaces in the pseudo-Euclidean space $\mathbb{E}^{n+1}_s$, which write as a warped product of a $1$-dimensional base with an $(n-1)$-manifold of constant sectional curvature. We show that either they have constant sectional curvature or they are contained in a rotational hypersurface. Therefore, we first define rotational hypersurfaces in the pseudo-Euclidean space.
In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces.
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of GCR hypersurfaces with vanishing Gauss-Kronecker curvature. We also give some explicit examples. Keywords: Generalized constant ratio submanifolds, $δ(r)$-invariant hypersurfaces, constant mean curvature, Gauss-Kronecker curvature
In this note we improve a gap result concerning the range of the mean curvature of complete $CMC$ proper-biharmonic hypersurfaces in unit Euclidean spheres.
In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.
We prove that strong finite total curvature complete hypersurfaces of (n+1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punctures.
The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.
In this paper, we characterize round spheres in the Euclidean space under some suitable conditions on the r-mean curvature.
We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.
We prove that there are not algebraic hypersurfaces of degree 3 in $\mathbb{R}^n$ with non zero constant mean curvature.
On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds
本文研究了极小与稳定极小超曲面的刻画问题，确立了其在 Euclid 空间中表现为超平面、在 Riemann 流形中表现为全测地子流形的条件。
In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either $\frac{n^2}{4}$ or $\frac{(n-2)^2}{4|x|^2}$. Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular domains with a conformally Euclidean metric.
We prove $ε$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $δ$-small compared to the mean curvature. We give the explicit dependence of $δ$ on $ε$ within the class of uniformly convex hypersurfaces with bounded volume.
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.
By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.
Euclidean hypersurfaces with semi-parallel Moebius second fundamental form
该论文对欧几里得空间中n≥4维的无脐超曲面，按半平行Moebius第二基本形式及三个不同主曲率的条件完成了分类。
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).
A Ricci soliton $(M,g,v,λ)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.
In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost Einstein is diffeomorpic and almost isometric to a standard sphere.
We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.
In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal null 2-type hypersurface in a Euclidean space has constant mean curvature and constant scalar curvature.
A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case of complete hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 4$. If there is no open subset where $f$ is either totally geodesic or a cylinder over an unbounded hypersurface of $\mathbb{R}^4$, we prove that $f$ is infinitesimally bendable only along ruled strips. In particul…
In the realm of conformal geometry, we give a classification of the Euclidean hypersurfaces that admit a non-trivial conformal infinitesimal variation. In the restricted case of conformal variations, such a classification was obtained by E. Cartan in 1917. The case of infinitesimal isometric variations was done by U. Sbrana in 1908. In particular, we show that the class of hypersurfaces that allow a conformal infinitesimal variation is much larger than the one considered by Cartan.
In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.
In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved by A. Fraser and M. Li \cite{FRLI}. As a consequence, we obtain the two-piece property for free boundary minimal hypersurfaces in the unit ball: every equatorial disk divides any compact minimal hypersurface with free boundary in the unit ball in two connected pieces.
We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with constant mean curvautre. Next, we prove the same kind of theorem for generalized rotational hypersurfaces of O(l+1) x O(m+1)-type. The key lemmas in this paper show the existence of solutions for singular initial value problems which arise from the analysis of ordinary differential equations of generating curves of those hypersurfaces.
In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or $λ_1$ have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on $v_M\|\B\|_α^n$ and show that their spectral and topological properties depends on the position of $α$ with respect to the critical value $\dim M$. The study of the metric shape of these extremal hypersurfaces will be done in \cite{AG1}, using estimates of the present paper.
In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
In this paper, we extend the notion of affine translation surfaces introduced by Liu and Yu (Proc. Japan Acad. Ser. A Math. Sci. 89, 111--113, 2013) in a Euclidean space R^{3} to higher dimensional ambient spaces. We provide that an affine translation hypersurface of constant Gauss-Kronocker curvature K_{0} in R^{n+1} is a cylinder, i.e. K_{0}=0. As further applications we describe such hypersurfaces in the isotropic spaces satisfying certain conditions on the isotropic curvatures and the Laplacian.
We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface $f\colon\, M^n\to\R^{n+1}$ means that $\hat f$ is nowhere a composition $\hat f=\hat F\circ f$, where $\hat F\colon\,V\subset \R^{n+1}\to\R^{n+2}$ is an isometric immersion of an open subset $V$ containing $f(M)$.
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