Closed Weingarten hypersurfaces in semi-riemannian manifolds

Closed Weingarten hypersurfaces in warped product manifolds

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $ψ$ in $\bar M$, we find a closed hypersurface $Σ$ which is solution of an equation of the form $F(B)=ψ$, where $B$ is the second fundamental form of $Σ$ and $F$ is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature.

The boundary value Minkowski problem for Weingarten curvatures

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are established under the natural assumptions that the prescribed boundary is strictly convex and the prescribed function satisfies a Serrin type condition.

Lipschitz Continuous Hypersurfaces with Prescribed Curvature and Asymptotic Boundary in Hyperbolic Space

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.

Prescribed Weingarten curvature equations in warped product manifolds

In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured.

规定曲率问题的凸毛细超曲面

Convex capillary hypersurfaces of prescribed curvature problem

本文将闭凸超曲面的规定$k$-阶Weingarten曲率问题扩展到毛细情形，通过将其转化为球冠上带Robin边界条件的Hessian商方程可解性问题，在自然充分条件下证明了严格凸毛细超曲面的存在性，扩展了毛细Minkowski问题的相关研究。

Self-Similar Solutions to Curvature Flow of Convex Hypersurfaces

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons w…

Lie 极小 Weingarten 曲面

Lie minimal Weingarten surfaces

本文研究黎曼空间形式中 Lie 球面不变能量的临界点 Lie 极小曲面，通过 Euler-Lagrange 方程刻画其主曲率，并证明常平均曲率曲面及平坦外围空间中满足仿射关系的曲面和椭圆线性 Weingarten 曲面均具有旋转性。

Weingarten Surfaces Associated to Laguerre Minimal Surfaces

In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we define for any surface $Σ$ its spherical mean curvature $H_S$ which depends on principal curvatures of $Σ$ and the radius function $R$. Then we consider two classes of surfaces: the ones with $H_S = 0$, called $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, named $H_2$-surfaces. We provide for each these classes a Weierstrass-typ…

Geometric Characterizations of Canal Hypersurfaces in Euclidean Spaces

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…

Curvature estimates for spacelike graphic hypersurfaces in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed $2$-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane $\mathscr{H}^{n}(1)\subset\mathbb{R}^{n+1}_{1}$ of center at origin and radius $1$, can be proven.

Isoperimetric and Weingarten surfaces in the Schwarzschild manifold

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric surfaces for very large enclosed volume. This complements work in H. Bray's thesis, where isoperimetric surfaces homologous to the horizon are studied.

Canal Hypersurfaces Generated by Non-Null Curves in Lorentz-Minkowski 4-Space

In the present paper, firstly we obtain the general expression of the canal hypersurfaces which are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hypercones in $E_{1}^{4}$ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in $E_{1}^{4}$ by taking constant radius function and finally…

双曲空间中的极限球面p-Christoffel-Minkowski问题与给定p-偏移Weingarten曲率问题

The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space

本文研究双曲空间中凸超曲面的对应几何问题，通过建立新满秩定理证明极限球面p-Christoffel-Minkowski问题存在光滑原点对称严格凸解，并提出给定p-偏移Weingarten曲率问题且证明其存在性结果。

On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one piecewise smooth horizontal curve in this hypersurface connecting them. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry. We use this …

A note on starshaped compact hypersurfaces with a prescribed scalar curvature in space forms

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $σ_2(κ(X))=f(X, ν(X)).$ In this note, we give a simpler proof of this result, and extend it to space forms.

关于黎曼流形中扭形成向量场与双调和超曲面

On torse-forming vector fields and biharmonic hypersurfaces in Riemannian manifolds

本文研究黎曼流形中具有扭形成向量场的双调和超曲面，并给出了其若干性质。

Biharmonic space-like hypersurfaces in pseudo-Riemannian space

We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.

Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups

We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.

Lightlike Hypersurfaces of Metallic Semi-Riemannian Manifolds

In our article, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces. We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hypersurfaces, investigate integrability conditions for the distributions and give some examples.

On biconservative hypersurfaces in 4-dimensional Riemannian space forms

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.

Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs

We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $ζ$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $ζ$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.

Some existence results to the Dirichlet problem for the minimal hypersurface equation on non mean convex domains of a Riemannian manifold

We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds

Riemannian超曲面总曲率的连续性

Continuity of total curvatures of Riemannian hypersurfaces

研究Riemannian流形中正达的超曲面及Cartan-Hadamard空间中的凸体的总广义均曲率在Hausdorff距离下的连续性。

Ricci Solitons on Pseudo-Riemannian Hypersurfaces of 4-dimensional Minkowski space

In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,λ)$ in both Riemannian and Lorentzian settings. First, we obtain the necessary and sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in $\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,λ)$. In each of the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain cha…

关于 Euclid 空间与 Riemann 流形中的极小超曲面

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。

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minim…

A Filtration for Isoparametric Hypersurfaces in Riemannian Manifolds

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,...,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces.

Closed hypersurfaces of prescribed mean curvature in locally conformally flat Riemannian manifolds

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.

Riemannian流形中极小嵌入超曲面第一特征值的下界

Lower bound for the first eigenvalue of a minimally embedded hypersurface in a Riemannian manifold

本文研究了Riemannian流形中极小嵌入超曲面的谱性质，在Ricci曲率具有正下界的紧致可定向流形条件下，给出了闭可定向超曲面上Laplace-Beltrami算子第一特征值的下界。

Null screen quasi-conformal hypersurfaces in semi-Riemannian manifolds and applications

We introduce a class of null hypersurfaces of a semi-Riemannian manifold, namely, screen quasi-conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.

The Dirichlet problem for the minimal hypersurface equation with Lipschitz continuous boundary data on domains of a Riemannian manifold

Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an extension to Riemannian manifolds of an existence result of G. H. Williams ( J. Reine Angew. Math. 354:123-140, 1984).

Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups

Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications are discussed.

Locally Product-like Statistical Manifolds and Their Hypersurfaces

In this paper, almost product-like Riemannian manifolds are investigated. Basic properties on tangential hypersurfaces of almost product-like Riemannian manifolds are obtained. Some examples of tangential hypersurfaces are presented. Some relations involving the Riemannian curvature tensor of a tangential hypersurface are computed.

紧致类空Cauchy超曲面空间的黎曼几何

The Riemannian geometry of the space of compact spacelike Cauchy hypersurfaces

研究全局双曲时空中紧致类空Cauchy超曲面空间上的弱黎曼度量，证明测地距离为正且截面曲率非正。

Eigenvalue estimates of minimal hypersurfaces with finite index in Riemannian manifolds

The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (\cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (\cite{SchoenYau1976}). Recently, Seo (\cite{Seo14}) generalized Theorem 1.7 (\cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (\cite{SchoenYau1976}) and 1.8 (\cite{Seo14})

Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.

Semi-Riemannian hypersurfaces in $\mathbb{L}^{n+1}$ with a totally geodesic foliation of codimension one

We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.

Strictly Locally Convex Hypersurfaces with Prescribed Curvature and Boundary in Space Forms

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.

Geometry of isoparametric hypersurfaces in Riemannian manifolds

In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of the Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian man…

Four dimensional hypersurfaces with proper mean curvature vector field in pseudo-Riemannian space forms

In this paper, we study four dimensional hypersurface M^4_r with proper mean curvature vector field (i.e. Δ\vec{H} is proportional to \vec{H}) in pseudo-Riemannian space form N^5_s(c), and show that it has constant mean curvature, and give the range of this constant. As an application, we get that biharmonic hypersurfaces in N^5_s(c) are minimal in some specific cases, which partially confirms B.-Y. Chen's conjecture.

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

In this paper, we show that, for a biharmonic hypersurface $(M,g)$ of a Riemannian manifold $(N,h)$ of non-positive Ricci curvature, if $\int_M|H|^2 v_g<\infty$, where $H$ is the mean curvature of $(M,g)$ in $(N,h)$, then $(M,g)$ is minimal in $(N,h)$. Thus, for a counter example $(M,g)$ in the case of hypersurfaces to the generalized Chen's conjecture (cf. Sect.1), it holds that $\int_M|H|^2 v_g=\infty$.

Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by $J$ be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field $N$ is said to be \em Hopf \em if the tangent vector field $JN$ is a principal direction. It …

On compact hypersurfaces in a Riemannian vector bundle with prescribed vertical Gaussian curvature

Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial graph over the unit sphere subbundle of E and prove its existence by solving in this one a nonlinear partial differential equation of Monge-Ampère type.

黎曼超曲面的总平均曲率比较公式

Comparison formulas for total mean curvatures of Riemannian hypersurfaces

本研究通过构造陈省身型微分形式，推导黎曼流形嵌套超曲面总平均曲率的比较公式，为研究者与Joel Spruck此前通过Reilly恒等式得到的相关结果提供了更简洁的证明方法。

伪黎曼空间形式中指标2的4维等参超曲面

4-Dimensional Isoparametric Hypersurfaces of Index 2 in the Pseudo-Riemannian Space Forms

本文研究伪黎曼空间形式中指标2的等参超曲面，基于Petrov分类给出分类，提供实例，并证明某些情况下不存在复主曲率的此类超曲面。

Riemannian流形中凸超曲面的最优下曲率界

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

研究证明了Riemannian流形中凸超曲面的曲率下界，优化了Buyalo定理的曲率下界。

Constant $k$-curvature hypersurfaces in Riemannian manifolds

Rugang Ye proved the existence of a family of constant mean curvature hypersurfaces in an $m+1$-dimensional Riemannian manifold $(M^{m+1},g)$, which concentrate at a point $p_0$ (which is required to be a nondegenerate critical point of the scalar curvature), moreover he proved that this family constitute a foliation of a neighborhood of $p_0$. In this paper we extend this result to the other curvatures (the $r$-th mean curvature for $1\le r\le m$).

A Survey on Convex Hypersurfaces of Riemannian Manifolds

We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.

A Classification of Riemannian manifolds of quasi-constant sectional curvatures

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean space, while the Riemannian QC-manifolds with negative horizontal sectional curvatures are locally equivalent to canal space-like hypersurfaces in Minkowski space. We prove that the local theory of conformally flat Riemannian manifolds, which can be locally isometrically embedded as…

关于Riemann流形中q-凸超曲面

On $q$-convex hypersurfaces in Riemannian manifolds

本文研究了Riemann流形中闭合凸超曲面的拓扑性质，证明在满足一定曲率条件时，此类超曲面是有限根本群的有理同调球体，同时给出了q-凸超曲面在平均曲率条件下的类似结论。

Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds for the initial data and the corresponding ambient space. As an application, we provide a class of Riemannian manifolds that admit hypersurfaces with constant principal curvatures, which are not isoparametric hypersurfaces. Furthermore, for a class of ambient spaces, we show that the singularities developed by the mean curvature flow with isoparametric hypersu…

高维黎曼流形中体积极小自由边界超曲面的刚性

Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds

本文针对高维黎曼流形中的体积极小自由边界超曲面，建立了基于数量曲率和边界第二基本形式的体积估计，并在等式情形下证明了流形的刚性。