Gregorio Pacelli Bessa, J. Fabio Montenegro
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 $.
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We find the necessary and sufficient condition for the proper biharmonic CMC hypersurfaces in the special warped product space. Furthermore, we obtain that there exists no proper biharmonic CMC compact hypersurface there.
It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
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.
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface M\subset S^{n+1} with cmc cannot take the values 1,2,3... n.
We prove that there are not algebraic hypersurfaces of degree 3 in $\mathbb{R}^n$ with non zero constant mean curvature.
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。
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.
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…
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.
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.
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.
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。
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 …
We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.
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。
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。
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…
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。
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。
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 $…
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。
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…
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.
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。
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.
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…
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。
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}.
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'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 $\mathbb{R}^4$、$\mathbb{R}^5$。
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…
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.
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.
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。
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。
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。
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。
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。
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。
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…
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.
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…
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".
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 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。
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。
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…
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…
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…
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…
We prove that globally subanalytic nonsingular CMC surfaces of $\mathbb{R}^3$ are only planes, round spheres or right circular cylinders
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。
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。
An Abresch-Rosenberg type tensor and a gap theorem for CMC hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$
该研究构造了S^n×R和H^n×R中CMC超曲面的Abresch-Rosenberg型对称无迹张量，结合积分Simons恒等式得到显式间隙定理，还确定了阈值的二阶大h渐近行为。
Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and w…
On isolated singularities and generic regularity of min-max CMC hypersurfaces
本文证明正里奇曲率流形中Allen-Cahn极小极大常平均曲率超曲面在孤立奇点处局部极小且切锥面积极小，并通过手术构造一般8维流形上的闭嵌入光滑常平均曲率超曲面。
Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces
本文针对维数4至7的无边界紧致黎曼流形中气泡收敛的极小或H-CMC超曲面序列，证明了指标加零化度的上半连续性，补充了已知的下半连续性结果，方法基于加权特征值分析与Lorentz--Sobolev不等式。
In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\R^{n}$ (Delaunay unduloids). When $n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space.
Suppose ${\cal L}$ is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature. We prove that every limit leaf of ${\cal L}$ is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of ${\cal L}$ has the structure of a lamination.
An extension of Liebmann's Theorem to hypersurfaces with boundary
本文将Liebmann定理推广至带边界超曲面，证明满足条件的CMC超曲面位于边界超平面半空间且继承边界对称性，得出边界为（n-1）球面且常平均曲率非零的此类超曲面仅为球冠的结论。
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