Curvature flows and CMC hypersurfaces

CMC hypersurfaces with bounded Morse index

We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. We also provide bounds on the area of separating CMC surfaces of bounded (Morse) index and use this, together with the previous results, to bound their genus.

Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^n (n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form $\mathbb{R}^{n+1}(c)$ with four distinct principal curvatures and the multiplicity of the zero principal curvature is at most one, then $M$ has constant scalar curvature. In particular, we obtain any CMC proper triharmonic hypersurface in $\mathbb{R}^5(c)$ is minimal when $c\le 0$, which supports the generalized Chen's conjecture. We also give some characterizations of CMC pr…

Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area minimizing, convex hull property, exchange roundoff trick to the CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.

Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spatial fiber of de Sitter Schwarzschild spacetime.

Deformations of free boundary CMC hypersurfaces

We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We discuss applications to free boundary CMC disks and Delaunay annuli in the unit ball of a space form.

奇异常平均曲率超曲面的等周问题

Isoperimetry of singular CMC hypersurfaces

该研究在黎曼流形中构造了首个非零常平均曲率的奇异等周边界，通过建立含特定奇点的常平均曲率超曲面的弱严格稳定性与局部带体积约束唯一极小化的关联，解决了相关等周问题。

Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures

A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $\mathbb{R}^{n+1}(c)$, then $M$ has constant scalar curvature. This supports the generalized Chen's conjecture when $c\le 0$. When $c=1$, we give an optimal upper bound of the mean curvature $H$ for a non-totally umbilical proper CMC $k$-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we …

变分Riemann度量下常均曲率超曲面的存在性

Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics

本文研究了在变化的Riemann度量下，常均曲率超曲面的存在性问题，证明了在给定度量下，若闭合c-CMC超曲面数量有限，则存在另一个度量使超曲面数量严格增加，并给出了度量差的L^{(n+1)/2}范数的精确上界。

旋转域中CMC自由边界超曲面的间隙结果与存在性

Gap results and existence of CMC free boundary hypersurfaces in rotational domains

本文研究旋转域中自由边界常平均曲率超曲面的存在性与唯一性，在特定条件下将其分类为拓扑圆盘或环面，并构造满足间隙条件的自由边界极小曲面例子。

On triharmonic hypersurfaces in space forms

In this paper we study triharmonic hypersurfaces immersed in a space form $N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the sphere $\mathbb S^{n+1}$ has constant scalar curvature; any CMC triharmonic hypersurface in the hyperbolic space $\mathbb H^{n+1}$ is minimal. Moreover, we show that any CMC triharmonic hypersurface in the Euclidean space $\mathbb R^{n+1}$ is minimal provided that the multiplicity of the principal curvature zero is at most one. In particular, we are able to prove that every CMC triharmonic hypersurface in the Euclidean space $\mathbb R^{6}$ is min…

具有小平均曲率的多重闭CMC超曲面的存在性

Existence of multiple closed CMC hypersurfaces with small mean curvature

该研究在闭黎曼流形上证明了小平均曲率闭CMC超曲面的多重存在性，给出了定量计数下界，推广了Zhou和Zhu关于单个闭c-CMC超曲面存在性的结果。

On compact CMC-Hypersurfaces of $N\times \mathbb{R}$

Let ${\mathscr F}(N\times \mathbb{R})$ be the set of all closed $H$-hypersurfaces $M\subset N\times \mathbb{R}$, where $N$ is a simply connected complete Riemannian $n$-manifold with sectional curvature $K_{N}\leq -κ^{2}<0$. We show that ${\Hm}(N\times \mathbb{R})=\inf_{M\in {\mathscr F}(N\times \mathbb{R})}\{| H_{M}| \}\geq (n-1)κ/n $.

空间形式中常平均曲率超曲面的局部刚性（II）

Local rigidity of constant mean curvature hypersurfaces in space forms (II)

本文研究空间形式中常平均曲率超曲面的局部刚性，针对不同主曲率个数 $g\geq 4$ 的情形，在平均曲率恒定且存在足够多重主曲率时，证明其为等参超曲面。

On the asymptotic Plateau problem for CMC hypersurfaces in hyperbolic space

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $Γ\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\} $ then, given $\left\vert H\right\vert <1,$ there is a complete, properly embedded, CMC $H$ hypersurface $Σ$ of $\mathbb{H}^{n+1}$ such that $\partial_{\infty }S=Γ\cup\left\{ x_{n+1}=+\infty\right\} .$ This result can be seen as a limit case of the existence theorem proved by B. Guan and J. Spruck in \cite{GS} on CMC $…

维数至少为四的每个双曲空间中强稳定的常平均曲率为一的超曲面

Strongly stable CMC-one hypersurfaces in every hyperbolic space of dimension at least four

本文证明在维数至少为四的双曲空间中，一类经典旋转超曲面具有强稳定性，并给出定量不等式，导致端点horospherical刚性失效。

On the asymptotic Plateau's problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symm…

Gauss map and the topology of constant mean curvature hypersurfaces of $\mathbb{S}^{7}$ and $\mathbb{CP}^{3}$

We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $\mathbb{CP}^{3}$ and obtain similar results for CMC hypersurfaces of this space.

5维和6维流形中常平均曲率超曲面的刚性

Rigidity of CMC hypersurfaces in 5-and 6-manifolds

本文证明在5维和6维流形中，非负中间里奇曲率等曲率条件下，稳定极小超曲面及非零平均曲率常平均曲率超曲面的刚性，推广了Chodosh-Li-Stryker的结果。

Cmc Doublings of Minimal Surfaces via Min-Max

Let $Σ^2 \subset M^3$ be a minimal surface of index 0 or 1. Assume that a neighborhood of $Σ$ can be foliated by constant mean curvature (cmc) hypersurfaces. We use min-max theory and the catenoid estimate to construct $\varepsilon$-cmc doublings of $Σ$ for small $\varepsilon > 0$. Such cmc doublings were previously constructed for minimal hypersurfaces $Σ^n \subset M^{n+1}$ with $n+1\ge 4$ by Pacard and Sun using gluing methods.

Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving an effective version of the compactness theorem for weakly stable CMC hypersurfaces established in the recent work of the first and third-named authors. Our results generalize the curvature estimate and the sheeting theorem proven respectively by Schoen--Simon--Yau and Schoen--Si…

双曲空间$\u210D^4$中具有有限指数的CMC超曲面

CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$

本文利用μ-bubble方法证明了双曲空间$\u210D^4$及满足曲率条件的4维黎曼流形中，不存在特定的完备非紧常平均曲率/极小稳定超曲面，推进了几何测度论相关不存在性问题的研究。

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}.

On stable CMC hypersurfaces with free-boundary in a Euclidean Ball

In this note, we observe that if $B$ is a ball in a Euclidean space with dimension $n$, $n\geq3$, then a stable CMC hypersurface $Σ$ with free boundary in $B$ satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where $L$, $A$ and $H$ denote the length of $\partial Σ$, the area of $Σ$ and the mean curvature of $Σ$, respectively. Consequently, if the boundary $\partial Σ$ is embedded then $Σ$ must be totally geodesic or starshaped with respect to the center of the ball. This result is an improvement of a theorem proved by A. Ros and E. Vergasta \cite{R-V} . In particu…

Do Carmo 问题在 $\mathbb{R}^6$ 中常均曲率超曲面上的研究

Do Carmo's problem for CMC hypersurfaces in $\mathbb{R}^6$

本文证明 $\mathbb{R}^6$ 中有限指标完备非紧常均曲率超曲面必为极小，肯定回答 do Carmo 问题，并推广至 $\mathbb{R}^4$、$\mathbb{R}^5$。

CMC hypersurfaces with two principal curvatures

This paper explains the construction of all hypersurfaces with constant mean curvature -- cmc -- and exactly two principal curvatures on any space form endowed with a semi-riemannian metric. Here we will consider riemannian hypersurfaces as well as hypersurfaces with semi-riemannian metrics induced by the ambient space. We show explicit immersions for all these hypersurfaces. We will see that for any given ambient space, the family of cmc hypersurfaces with two principal curvatures depends on two parameters, $H$ and $C$. We describe the range for all possible $(H,C)$ associated with complete c…

Spacelike spherically symmetric CMC hypersurfaces in Schwarzschild spacetimes (I): Construction

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.

Min-max theory for constant mean curvature hypersurfaces

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover, if $c$ is nonzero then our min-max solution always has multiplicity one.

具有非正 Yamabe 不变量的毛细 cmc 超曲面的面积估计

Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant

本文研究具有非正 Yamabe 不变量的稳定毛细 cmc 超曲面，在数量曲率和边界平均曲率有下界的流形中建立面积估计，并对嵌入且 $\mathcal{J}$ 能量极小的情形给出局部分裂刚性结论。

半空间中奇异毛细管CMC超曲面的Alexandrov型定理

Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space

本文研究了半空间中相对等周型问题临界点的分类，通过引入新型移位距离函数，在特定正则性假设下证明了奇异毛细管CMC超曲面的Alexandrov型定理。

二维、三维和四维近稳定 H-超曲面的半径估计

Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4

本文将二维至四维完备常平均曲率超曲面的半径估计从稳定情形推广到近稳定情形，并证明某些有效嵌入的CMC超曲面必为proper嵌入。

关于上半球中自由边界CMC超曲面的Morse指数

On the Morse index of free-boundary CMC hypersurfaces in the upper hemisphere

该文研究上单位半球中自由边界CMC超曲面的Morse指数与Jacobi算子第一特征值，给出不同几何类型超曲面的指数取值、特征值估计及等号成立条件，推进了微分几何中相关问题的研究。

有限指数常均曲率超曲面在低维空间中的研究

Finite index constant mean curvature hypersurfaces in low dimensions

本文证明在六维流形中，完备有限指数浸入的常均曲率超曲面要么是极小的要么是紧致的，扩展了低维结果，并完成六维正曲率空间中双侧完备弱稳定常均曲率超曲面的分类。

探索球面中紧致常平均曲率超曲面的空间，第二部分

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

本文借鉴 R^3 中 Delaunay CMC 曲面的构造，在 S^4 中粘合全脐球面与 Clifford 超曲面，并通过数值去奇异化得到嵌入与非嵌入两类 CMC 超曲面族。

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…

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.

类时CMC曲面上的Hopf微分与曲率线流

Hopf differentials and curvature line flows on time-like CMC surfaces

该研究探讨洛伦兹3空间形式中类时CMC曲面上Hopf微分与曲率线流的关联，证明非退化Hopf微分下脐点处曲率线流的指标由其阶数模4的余数确定，丰富了类时曲面的几何性质研究。

Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. O…

The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups

In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the (2m+1)-dimensional Heisenberg group we also derive necessary and sufficient conditions for the Gauss map to be harmonic and prove that for m=1 all CMC-surfaces with the harmonic Gauss map are "cylinders".

CMC hypersurfaces of semi-Riemannian groups

In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface $M$ of a semi-Riemannian group $G$ of Lie algebra $\mathfrak g$ and index 0 or 1. If $G$ is Riemannian and $M$ is compact and transversal to an element of $\mathfrak g$, we show that it is a lateral class of a closed embedded Lie subgroup of $G$; we also do this if $G$ is Lorentzian, provided $M$ has sufficiently large mean curvature. If $G$ is Riemannian semisimple and $M$ is compact, we prove that $M$ has degenerate Gauss map and minimal relative nullity at least 1. We also extend the above results to th…

CMC proper-biharmonic surfaces of constant Gaussian curvature in spheres

CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in $\sn^5$.

欧几里得空间上的CMC叶状结构是极小叶状结构

CMC foliations on Euclidean spaces are minimal foliations

该研究解决了微分几何中叶状结构领域的经典猜想，证明$\boldsymbol{\text{R}}^{n+1}$上由CMC超曲面构成的叶状结构是极小叶状结构，且当$n \boldsymbol{\text{≤ 6}}$时此类叶状结构由平行超平面组成。

Minkowski (n+1) 空间中的完备 CMC 超曲面

Complete CMC hypersurfaces in Minkowski (n+1)-space

本文证明 Minkowski 空间任意正则区域存在唯一的类空 CMC 超曲面叶层结构，完成整类空 CMC 超曲面分类，并据此得到 2+1 维常 Gauss 曲率整曲面与双曲平面直凸区域等距。

CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere

CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case.

CMC hypersurfaces on riemannian and semi-riemannian manifolds

In this paper we show explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-riemannian manifolds with constant sectional curvature. In particular, we prove that every h in [-1,-2 sqrt{n-1}/n) can be realized as the constant curvature of a complete immersion of S_1^{n-1} x R in the (n+1)-dimensional de Sitter space S_1^{n+1}. We provide 3 types of immersions with cmc in the Minkowski space, 5 types of immersion with cmc in the de Sitter space and 5 types of immersion with cmc in the anti de Sitter space. At the end of the…

Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Blocks

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These `balancing conditions' are analogous to those that must be satisfied in the `classical' context of gluing constructions of CMC hyp…

Cosmological time versus CMC time I: Flat spacetimes

This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they are causally incomplete. The proof, which is based on using the level sets of the cosmological time function as barriers, is conceptually simple and will provide the basis for future work on constant mean curvature time functions in general constant curvature spacetimes, as well f…

Stable hypersurfaces with constant higher order mean curvature

We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for CMC capillary hypersurfaces.

Cosmological time versus CMC time II: the de Sitter and anti-de Sitter cases

This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cases of negative curvature (i.e. anti-de Sitter) spacetimes and postitive curvature (i.e. de Sitter) spacetimes is dealt with. As in the flat case, the existence of CMC time functions is obtained by using the level sets of the cosmological time function as barriers. A major part of…

Hypersurfaces with a canonical principal direction

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the further condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We a…

Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature I

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. In this paper, we completely classify CMC-1 surfaces with dual total absolute curvature at most 4π. Moreover, we give new examples and partially classify CMC-1 surfaces with dual total absolute curvature at m…

Bifurcation of constant mean curvature tori in Euclidean spheres

We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

关于$\boldsymbol{\text{S}^5(1)}$中具有常数量曲率的闭CMC超曲面的刚性

On the Rigidity of Closed CMC Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar Curvature

该研究证明$\boldsymbol{\text{S}^5(1)}$中具有常数量曲率的闭CMC超曲面若某点仅两个主曲率则为等参Clifford环面，加Willmore条件后可完整分类为几类等参超曲面，采用无迹张量恒等式等方法完成证明。

CMC foliations of closed manifolds

We prove that every closed, smooth $n$-manifold $X$ admits a Riemannian metric together with a smooth, transversely oriented CMC foliation if and only if its Euler characteristic is zero, where by CMC foliation we mean a codimension-one, transversely oriented foliation with leaves of constant mean curvature and where the value of the constant mean curvature can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of $X$ can be chosen so that the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal …

球面中的闭双保守超曲面

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。