Thomas A. Ivey, Patrick J. Ryan
We update and refine the work of T. Hamada concerning *-Einstein hypersurfaces in complex space forms CP^n and CH^n. We also address existence questions using the methods of moving frames and exterior differential systems.
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Einstein hypersurfaces are ``very rare" in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
The non-existence of three dimensional real hypersurfaces in non-flat complex space forms with parallel *-Ricci tensor is proved.At the end of the papaer ideas for further research on *-Ricci tensor are provided.
Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$. Suppose that the structure vector field $ξ$ of $M$ is an eigen vector field of the Ricci tensor $S$, $Sξ=βξ$, $β$ being a function. We study on $M$, a gradient pseudo-Ricci soliton as an extended concept of Ricci soliton, closely related to pseudo-Einstein real hypersurfaces. We show that a $3$-dimensional ruled real hypersurface of $M^2(c), c<0$ admits a non-trivial gradient pseudo-Ricci soliton.
This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case of ambient space being the complex hyperbolic space are provided. Furthermore, new results concerning $ξ$-parallelism of $^{*}$-Ricci tensor of real hypersurfaces in non-flat complex space forms are presented.
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.
We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal or $\frak A$-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with commuting Ricci tensor.
Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex space form admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.
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 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…
In this paper, we have introduced a new notion of generalized Tanaka-Webster Reeb recurrent Ricci tensor in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Next, we give a non-existence property for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with such a condition.
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_2({\mathbb C}^{m+2})$. By using such parallel conditions, we give complete classifications of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.
In this article, we study the biharmonic hypersurfaces in the Sasakian space form with the induced metric of tensor Ricci. We find the existence necessary and sufficient condition of the biharmonic hypersurfaces there. We show that the biharmonic Hopf hypersurfaces are minimal where gradient of the mean curvature is in direction of the structural vector fields. Furthermore, we prove that does not exist any biharmonic Hopf hypersurfaces when the gradient of the mean curvature is a principal direction.
We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the background manifold is sufficiently pinched and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.
In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a consta…
We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.
Stability of weighted minimal hypersurfaces under a lower $1$-weighted Ricci curvature bound
本文研究了在下界1加权里奇曲率限制下加权极小超曲面的稳定性，推导了相关几何结论，并得出三维加权流形的结构定理及非存在性结果。
In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for $f$-stability index of a constant weighted mean curvature hypersurface with finite weighted volume and isometrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field globally defined. For such hypersurface, we still give a necessary condition for equality to be achieved in the estimate obtained.
We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.
Let $M^{n+1}$ be an orientable compact Riemannian manifold with positive Ricci curvature. We prove that the Almgren-Pitts width of $M^{n+1}$ is achieved by an orientable index $1$ minimal hypersurface with multiplicity $1$ and optimal regularity. This extends to dimensions $n+1\geq 8$ the results of Ketover-Marques-Neves arXiv:1601.04514v1 [math.DG] and X. Zhou arXiv:1504.00966v2 [math.DG].
On Ricci solitons whose level hypersurfaces have parallel second fundamental form
该研究刻画了水平超曲面具平行第二基本形式的梯度Ricci孤立子的局部结构，证明相关刚性结论并验证Cao猜想在特定情形下成立。
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 $\int_M|H|^2 v_g=\infty$.
Ricci pinched compact hypersurfaces in spheres
本文研究球面中 Ricci 曲率受平均曲率约束的紧致超曲面的拓扑，利用 Bochner 技巧获得比任意余维情形更强的结果。
We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurf…
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。
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.
Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms
本文证明实空间形式中连通超曲面具有常Ricci特征值当且仅当曲率齐性，从而完成分类，推广了Einstein超曲面的分类，并得到曲率非齐性流形不能等距浸入余维一实空间形式的推论。
Minimal hypersurfaces in manifolds of Ricci curvature bounded below
本文利用Colding方法研究Ricci曲率下有界流形中极小超曲面距离函数的角度估计，结合Cheeger-Colding理论获得Laplacian比较，并证明度量锥极限下截面的Frankel性质。
Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces
本文在Damek-Ricci空间中通过过度膨胀超球面构造新的球状等参超曲面族，连接测地球面与已知等参超曲面，并研究其齐次性和焦簇的全测地条件。
A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds
本文研究完备非紧 Riemann 流形有界区域的几何不等式，在渐近或积分 Ricci 曲率界条件下建立 Willmore 型不等式，并推广恢复 Jin-Yin 的近期结果。
Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below
本文在Ricci曲率有下界的完备流形中建立闭超曲面的Willmore型不等式，推广经典结果至负曲率情形，并应用于双曲空间得到测地球面的刻画。
Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces …
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. As an essential technical tool, we pr…
We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism between the fundamental groups of S and N induced by the inclusion is surjective, provided only that N - S has two connected components, one of which has noncompact closure and trivial homotopy relative to S. The idea of the proof is to view N as a spacelike hypersurface in a suitable Lorentz manifold and then apply a recent version of certain classic results b…
In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for th…
In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$ with commuting Ricci tensor. Each can be described as a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a h…
Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = ϕ^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded minimal hypersurface in $(M,g)$ of volume bounded by $C V^{\frac{n-1}{n}}$, where $V$ is the total volume of $(M,g)$ and $C$ is a constant that depends only on $n$. When $Ric(M,g_0) \geq -(n-1)$ we obtain a similar bound with constant $C$ depending only on $n$ and the volume of $(M,g_0)$. Our second result concerns manifolds $(M,g)$ of positive Ricci curvature. W…
An eigenvalue estimate for self-shrinkers in a Ricci shirinker
本文研究Ricci收缩孤立子中超曲面上漂移拉普拉斯算子的谱性质，证明了有界加权平均曲率浸入超曲面的谱离散性，并给出嵌入f-极小超曲面第一非零特征值的下界，推广了Choi-Wang等经典估计。
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。
We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension $3\leq n+1\leq 7$. Furthermore, we show that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index+area. When combined with the recent work fo F.C. Marques and A. Neves, we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found.
Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature
本文研究正Ricci曲率$G$-流形中的等变极小极大超曲面，证明其为指标1的重数1极小超曲面，并在三维保定向作用下得到亏格、奇点及$G$-宽度的相关界。
The main objective of this paper is to control the geometry of null cones with time foliation in Einstein vacuum spacetime under the assumptions of small curvature flux and a weaker condition on the deformation tensor for $\bT$. We establish a series of estimates on Ricci coefficients, which plays a crucial role to prove the improved breakdown criterion in [12].
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
Ricci curvature and minimal hypersurfaces with large Betti numbers
研究构造了高维里齐曲率正的闭合流形，其嵌入的双侧极小超曲面具有Morse指数为一且贝蒤数不一致有界的特性。
Willmore-type inequalities for closed hypersurfaces in weighted manifolds
该研究针对具非负Bakry-Émery里奇曲率的加权流形闭超曲面，推导稳态与收缩梯度Ricci孤立子中的精确Willmore型不等式并刻画等号情形，得到相关经典不等式的加权版本，还推出加权等周型不等式。
Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $Σ$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $Σ$ does not separate $M$ then $Σ$ is totally geodesic and $M\setminusΣ$ is isometric to the Riemannian product $Σ\times(a,b)$, and if $Σ$ separates $M$ then the map $i_*:π_1(Σ)\rightarrow π_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\geq(n-1)\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\geq-(n-1)k,k>0$, and for a free boundar…
ACS Condition on Minimal Isoparametric Hypersurfaces of Positive Ricci Curvature in Unit Spheres
本文研究了正各向同性曲率的最小等参超曲面的ACS条件，证明了当主曲率有4种不同取值时，ACS不等式成立当且仅当最小的乘数大于等于4。
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
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。
Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below
本文利用Cheeger-Colding理论，研究Ricci曲率下有界流形中面积极小超曲面在Gromov-Hausdorff收敛下的体积连续性，并通过blowing up论证给出极限超曲面奇异集的精确维数估计。
We prove a Bakry-Émery generalization of a theorem of Petersen and Wilhelm, itself a generalization of a theorem of Frankel, that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We then prove splitting theorems of Croke-Kleiner type for manifolds bounded by hypersurfaces obeying Bakry-Émery curvature bounds. Motivated in part by the near-horizon geometry programme of general relativity, we do not assume that the Bakry-Émery vector field is of gradient type.
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Ricci tensor is parallel with respect to the generalized Tanaka-Webster connection.
In this paper we study real hypersurfaces in the complex quadric space $Q^m$ whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature $α$ of such hypersurfaces is constant and if $α$ is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\mathbb{C} P^k \subset Q^m$, where $m=2k$. We also consider Reeb flat hypersurfaces, namely, when the Reeb curvature is zero. We show that the tube around $\mathbb{C} P^k \subset Q^m$ ($m=2k$), with radius $\fracπ{4}$ is the only Reeb flat Hopf hypersurface with commuting Ricci tens…
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 we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is isometric to a truncated cone over $\partialΩ$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version …
We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.
The structure and rigidiy of FBCMC hypersurfaces
本文研究FBCMC超曲面的结构与刚性，利用k-tri-Ricci和3-中间Ricci曲率条件证明5维流形中稳定自由边界极小超曲面的刚性，并刻画其单端性。
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