The First Eigenvalue for Compact Minimal Embedded Hypersurface in $S^{n+1}(1)$ ($n\geq 3$) is $n$

通过等参超曲面构造紧致嵌入的$\lambda$-超曲面

Construction of compact embedded $λ$-hypersurfaces via isoparametric hypersurfaces

本文构造了任意非负λ下拓扑为S^1×M的闭嵌入λ-超曲面，其中M为等参超曲面，推广了先前结果并补全了Riedler构造的缺失情形。

Embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m>0$ in $S^{n+1}(1)$, for $1\leq m\leq n-1$. In particular, if the $4^{\text{th}}$ mean curvature $H_4$ takes value between $\dfrac{1}{(\tan \fracπ{k})^4}$ and $\dfrac{k^4-4}{n(n-4)}$ for any integer $k\geq3$, then there exists an $n$-dimensional ($n\geq 5$) compact nontrivial embedded hypersurface with constant $H_4$ in $S^{n+1}(1)$.

单位球面中嵌入极小超曲面的第一特征值

On the First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere

针对单位球面中闭嵌入极小超曲面，改进了其Laplace-Beltrami算子第一非零特征值的下界，结果略优于Duncan-Sire-Spruck的界。

The Existence of Embedded $G$-Invariant Minimal Hypersurface

For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\R^{n+1}$ with $H^F_r={constant}$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.

单位球面中嵌入极小超曲面的第一特征值 I：Yau 猜想

The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture

本文证明单位球面中嵌入极小超曲面的第一非零特征值等于其维数，从而肯定解决 Yau 猜想，并由此推导出多个刚性定理。

嵌入的圆柱状与甜甜圈形 $\lambda$-超曲面

Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces

本文构造了圆柱状非凸完备嵌入 $\lambda$-超曲面以及两个不等距的紧致甜甜圈形嵌入 $\lambda$-超曲面，说明 $\lambda$-超曲面不满足自收缩子的平面区域猜想相关结论。

Embeddedness of least area minimal hypersurfaces

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded. As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embe…

Classification theorems for biharmonic real hypersurfaces in a complex projective space

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.

Embedded constant $m^{\text{th}}$ mean curvature hypersurfaces on spheres

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between $\dfrac{1}{(\tan \fracπ{k})^m}$ and $\frac{k^2-2}{n}(\frac{k^2+m-2}{n-m})^{\frac{m-2}{2}}$ for $1\leq m\leq n-1$ and any integer $k\geq 2$, then there exists at least one $n$-dimensional compact nontrivial embedded hypersurface with constant $H_m>0$ in $S^{n+1}(1)$. When $m=1$, our results reduce to the results of Perdomo \cite{[P]}; when $m=2$ and $m=4$, our results reduce to the re…

Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow

Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing this $G$-flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex s…

探索球面中紧致常平均曲率超曲面的空间，第一部分

Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I

本文构造并考察单位球面中的嵌入CMC超曲面族，为新实例提供依据，并针对$H=0$情形验证Yau关于Clifford超曲面面积最小的猜想。

Sphere-foliated minimal and constant mean curvature hypersurfaces in product spaces

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric.

$\boldsymbol{\rm \bb S}^{n+1}_+$中紧嵌入超曲面的一个同余定理

A congruence theorem for compact embedded hypersurfaces in $\mathbb{S}^{n+1}_+$

该研究借助空间形式的Reilly型公式，针对球面空间中可平均凸等距嵌入到上半球面的紧子流形，建立了余维数约化定理与同余定理，推进了微分几何中子流形刚性问题的研究。

Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel structure Jacobi operator

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

嵌入超曲面在正里奇曲率流形中的第一个特征值估计

A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds

该研究在正里奇曲率流形中，通过曲率界、法向注入半径和超曲面几何特性，推导出嵌入超曲面第一个非零特征值的下界估计，扩展了Choi和Wang的经典结果到非极小情况。

Min-max theory for free boundary minimal hypersurfaces I - regularity theory

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative R…

Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that, unlike the isotropic case, there exists an anisotropic energy density function such that there exist closed embedded equilibrium surfaces with genus zero in the three-dimensional Euclidean space each of which is not (any homothety and translation of) the …

Embedded constant mean curvature hypersurfaces on spheres

Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\fracπ{m}) and b_{m,n}=\frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}, then, there exists a non isoparametric, compact embedded hypersurface in S^{n+1} with constant mean curvature H that admits the group O(n)x Z_m in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Z_m are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made …

The existence of embedded minimal hypersurfaces

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to any $n$.

Toward a Classification of Conformal Hypersurface Invariants

Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a Riemannian (or Lorentzian) conformal manifold. We construct a finite and minimal family of hypersurface tensors -- the curvatures intrinsic to the hypersurface and the so-called ``conformal fundamental forms'' -- that can be used to construct natural conformal invariants of the hypersurface embedding up to a fixed order in hypersurface-orthogonal derivatives o…

Min-max minimal hypersurface in manifolds with convex boundary and $Ric\geq 0$

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.

On embedded minimal hypersurfaces in $S^{n+1}$ with symmetries

In this note, we generalize a characterization of the Clifford torus due to Ros. Let $f:M\rightarrow S^{n+1}$ be an embedded closed minimal hypersurface. Assume there are $(n+2)$ great hyperspheres of $S^{n+1}$ perpendicular to each other, such that $M$ is symmetric with respect to them. Let $S$ denote the square of the length of the second fundamental form of $f$ and let $\bar S=\frac{1}{Vol(M)}\int_{M} Sd M$ be the average of $S$. Then $\bar S\geq n$ with equality holding if and only if $f$ is the Clifford torus $C_{m,n-m}$. It can be rewritten as a Simons' type theorem: If $0\leq \int_M (n-…

A remark on compact hypersurfaces with constant mean curvature in space forms

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.

New examples of constant mean curvature hypersurfaces in the sphere

In this paper, firstly, we show the existence of a compact embedded constant mean curvature (CMC) hypersurface $Σ_1$ in $\mathbb{S}^{2n}$ of the type $S^{n-1} \times S^{n-1} \times S^{1}$. Moreover, the hypersurface $Σ_1$ exhibits $O(n)\times O(n)$ symmetry. Secondly, we show that there exists a compact embedded CMC-hypersurface $Σ_2 \subset \mathbb{S}^{3n-1}$ of the type $S^{n-1} \times S^{n-1} \times S^{n-1} \times S^{1}$. These results generalize the results of Carlotto and Schulz.

Min-max for phase transitions and the existence of embedded minimal hypersurfaces

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren-Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min-max theories.

Embedded cmc hypersurfaces on hyperbolic spaces

In this paper we will prove that for every integer n>1, there exists a real number H_0<-1 such that every H\in (-\infty,H_0) can be realized as the mean curvature of a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}. For $n=2$ we explicitly compute the value H_0. For a general value n, we provide function ξ_n defined on (-\infty,-1), which is easy to compute numerically, such that, if ξ_n(H)>-2π, then, H can be realized as the mean curvature of a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}.

Quaternionic-contact hypersurfaces

We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.

实射影空间中极小超曲面的指标

Index of minimal hypersurfaces in real projective spaces

本文研究实射影空间中极小超曲面的 Morse 指标，证明嵌入不稳定单侧极小超曲面的指标下界为 n+2 且由三次等参极小超曲面达到，并借助 Lawson 曲面证明三维情形存在任意奇数指标的闭嵌入双侧极小曲面。

Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $λ$, produces an embedded hypersurface of constant mean curvature $λ$ ($λ$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the fir…

自由边界G-不变极小超曲面的极小极大理论

Min-max theory for free boundary G-invariant minimal hypersurfaces

本文将自由边界极小极大理论推广到等变情形，证明余齐性≥3的紧李群等距作用下的紧黎曼流形中存在非平凡G-不变带自由边界极小超曲面，且在里奇曲率非负、边界严格凸时有无穷多此类超曲面。

Constructions of $H_r$-hypersurfaces, barriers and Alexandrov Theorem in $H^n \times R$

In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r$-graph for each value $0<H_r\leq\frac{n-r}{n}$. Also, for each value $H_r>\frac{n-r}{n}$, there is a unique embedded compact strictly convex rotational $H_r$-hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an …

A Möbius scalar curvature rigidity on compact conformally flat hypersurfaces in $\mathbb{S}^{n+1}$

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group of $\mathbb{S}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant Möbius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $$\mathbb{ S}^1(\sqrt{1-r^2})…

Hypersurfaces with constant sectional curvature of $\Sf^n\times \R$ and $\Hy^n\times \R$

We classify the hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$ with constant sectional curvature and dimension $n\geq 3$.

Minimal hypersurfaces with arbitrarily large area

For 3 $\leq$ n $\leq$ 7, we prove that a bumpy closed Riemannian n-manifold contains a sequence of connected embedded closed minimal surfaces with unbounded area.

光滑Dupin超曲面的解析性与紧致性

Analyticity and tautness of smooth Dupin hypersurfaces

本文证明满足Dupin条件的光滑浸入欧氏超曲面局部实解析且半代数，并由此推出紧致嵌入超曲面具有紧致性。

Constant mean curvature hypersurfaces with single valued projections on planar domains

A classical problem in constant mean curvature hypersurface theory is, for given $H\geq 0$, to determine whether a compact submanifold $Γ^{n-1}$ of codimension two in Euclidean space $\R_+^{n+1}$, having a single valued orthogonal projection on $\R^n$, is the boundary of a graph with constant mean curvature $H$ over a domain in $\R^n$. A well known result of Serrin gives a sufficient condition, namely, $Γ$ is contained in a right cylinder $C$ orthogonal to $\R^n$ with inner mean curvature $H_C\geq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $Γ…

Riemann minimal surfaces in higher dimensions

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.

The Existence of G-Invariant constant mean curvature Hypersurfaces

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits $M\setminus M^{reg}$ is a smooth embedded submanifold of $M$ without boundary and ${\rm dim}(M\setminus M^{reg})\leq n-2 $. Then for any $c\in\mathbb{R}$, we show the existence of a nontrivial, smooth, closed, $G$-equivariant almost embedded $G$-invariant hypersurface $Σ^n$ of constant mean curvature $c$.

An immersed $S^{n}$ $λ$-hypersurface

In this paper, we construct an immersed, non-embedded $S^{n}$ $λ$-hypersurface in Euclidean spaces $\mathbb{R}^{n+1}$.

Complete minimal hypersurfaces in quaternionic hyperbolic space

We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.

On Stable Hypersurfaces with Vanishing Scalar Curvature

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. Our second result is the Bernstein type one \emph{there is no entire graphs of $\mathbb{R}^4$ with zero scalar curvature such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere}. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and $\dfra…

Blaschke's problem for hypersurfaces

We solve Blaschke's problem for hypersurfaces of dimension $n\geq 3$. Namely, we determine all pairs of Euclidean hypersurfaces $f,\tilde{f}\colon M^n\to\R^{n+1}$ that induce conformal metrics on $M^n$ and envelope a common sphere congruence in $\R^{n+1}$.

Dirac Operator on Embedded Hypersurfaces

New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.

Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau's conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.

Minimal hypersurfaces with bounded index

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follows our local picture.

Minimal hypersurfaces in R^n \times S^m

We classify minimal hypersurfaces in $R^n \times S^m$, $n,m \geq 2$, which are invariant by the canonical action of $O(n) \times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.

Global Structure of Locally Convex Hypersurfaces in Finsler-Hadamard Manifolds

Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures

\(\mathbb{R}^4\) 中限制在空间内的极小超曲面的拓扑伯恩斯坦定理

Topological Half-Space Theorems for Minimal Hypersurfaces in $\mathbb{R}^4$

研究 \(\mathbb{R}^4\) 中极小超曲面的拓扑伯恩斯坦定理，证明曲率有界、与 \(\mathbb{R}^3\) 微分同胚且在平板中的完备恰当嵌入极小超曲面必为超平面，立方体积增长假设下半空间中的也成立。

A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $δ(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$, $$\int_{M}S \geq δ(n){\rm Vol}(M^n),$$ where $S$ is the squared length of the second fundamental form of $M^n$. The Perdomo Conjecture asserts that $δ(n)=n$ which is still open in general. As byproducts, we also obtain some integral inequalities and Simons-type pinching results on closed embedded (or immersed) minimal hypersurfaces, with the first positive eigenvalue $λ_1(…

关于收缩梯度Ricci孤子中极小超曲面的注记

Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

本文证明在标量曲率满足R≥(n-1)λ的收缩梯度Ricci孤子中，任何紧致双侧光滑稳定极小超曲面的第二基本形式和法向Ricci曲率必须为零，且面积极小超曲面的存在会导致流形分裂。

具有半平行Moebius第二基本形式的欧几里得超曲面

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。

On Pseudo-Einstein Real Hypersurfaces

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a constant, if and only if $M$ is a pseudo-Einstein real hypersurface.

球面中的德劳内超曲面

Delaunay hypersurfaces in spheres

本研究针对n≥3的球面中的Delaunay超曲面展开，补充了其分类中缺失的花型，并发现了该球面内具有非零常平均曲率的嵌入Delaunay超曲面。