Given any nondegenerate k-dimensional minimal submanifold K of codimension greater than 1, we prove the existence of families of constant mean curvature submanifolds, with mean curvature varying from one member of the family to another, which `condense' to K. In particular, our result proves the existence of constant mean curvature hypersurfaces with nontrivial topology in any Riemannian manifold.
We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out.
Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legendrian singularities.
The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $\mathcal{A}$-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with $g\geq3$ are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.
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…
Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is defined as the $k$-th power sum of the principal curvatures, or equivalently, of the shape operator. Many necessary restrictions involving principal curvatures, higher order mean curvatures and Jacobi operators on such submanifolds are obtained, which, among other things, generalize some classical results in the theory of isoparametric hypersurfaces given by E. …
The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian submanifold of $Q^n$ is locally the image under the Gauss map of several hypersurfaces of $S^{n+1}(1)$. In this paper, we give explicit constructions for these correspondences and we prove a relation between the principal curvatures of a hypersurface of $S^{n+1}(1)$ and the local angle functions of the corresponding Lagrangian submanifold of $Q^n$. The existence of such a relation is remarkable since the definition of the angle functions depen…
An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $ρ^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of…
We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.
We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric $g$ is replaced with $g'$ in a neighbourhood of $g$. Our examples $(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups $SU(n)$ for $n\leq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians.
Consider a codimension $1$ submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset\mathbb{R}^{n+2}$ is a hypersurface. The envelope of tangent spaces of $M$ along $N$ generalizes the concept of tangent developable surface of a surface along a curve. In this paper, we study the singularities of these envelopes. There are some important examples of submanifolds that admit a vector field tangent to $M$ and transversal to $N$ whose derivative in any direction of $N$ is contained in $N$. When this is the case, one can construct transversal plane bundles and affine metrics on $N$ with the desirable …
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifol…
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal null 2-type hypersurface in a Euclidean space has constant mean curvature and constant scalar curvature.
We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. Various classification results in the Cayley projective and hyperbolic planes and in complex two-plane Grassmannians are obtained under some additional assumptions.
Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.
We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used to define local angle functions on any Lagrangian submanifold of ${Q^*}^n$. We prove that a Lagrangian immersion into ${Q^*}^n$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagran…
In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using the concept of f-biharmonic maps and prove that the set of all f-biharmonic maps from 2-dimensional domain is invariant under the conformal change of the metric on the domain. We give an improved equation for f-biharmonic hypersurfaces and use it to prove some rigidity theorems a…
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.
The objet of this paper is the study of the variations of a functional whose integrant is the r-th weighted curvature on the hypersurface of a closed Riemannian manifold. Some applications to hypersurfaces of the Euclidean space and the unit round sphère are given.
Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are "Weyl" connections adapted to the induced structure on the hypersurface. We begin the study of these with their holonomy.
We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.
By using T. Oprea's optimization methods on submanifolds, we give another proof of the inequalities relating the normalized $δ-$Casorati curvature $\hatδ_c(n-1)$ for submanifolds in real space forms. Also, inequalities relating the normalized $δ-$Casorati curvature $δ_C(n-1)$ for submanifolds in real space forms are obtained. Besides, we characterize a kind of Casorati ideal hypersurface of Euclidean 4-space. We also show that this kind of Casorati ideal hypersurface is rigid.
In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be also considered as the extension of the classification of the conformal isotropic submanifolds in the conformal space ${\mathbb Q}^n_s$.
Given a domain $Ω$ of $\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\pa Ω$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial Ω$ perpendicularly.
A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically eq…
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern's conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.
The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by usi…
The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented $2$-planes in $\mathbb{R}^{n+2}$. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.
This paper is a continuation of [TY12] and [QTY13]. We show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore.
A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that {\it any biharmonic submanifold in a Euclidean space is minimal}. In the case of a hypersurface $M^n$ in $\mathbb R^{n+1}$, Chen's conjecture was settled in the case of $n=2$ by Chen and Jiang around 1987 independently. Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with $n=3$. However, the general Chen's conjecture on a hypersurface $M^n$ remains open for $n> 3$. In this paper, we settle Chen's conjecture for hypersurfaces in $\mathbb R^{5}$ for $n=4$.
Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such in…
We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifo…
A biconservative submanifold of a Riemannian manifold is a sub- manifold with divergence free stress-energy tensor with respect to bienergy. These are generalizations of biharamonic submanifolds. In 2013, B. Y. Chen and M.I. Munteanu proved that $δ(2)$-ideal and $δ(3)$-ideal biharmonic hypersurfaces in Euclidean space are minimal. In this paper, we generalize this result for $δ(2)$-ideal and $δ(3)$-ideal bisonservative hypersurfaces in Euclidean space. Also, we study $δ(4)$-ideal biconservative hypersurfaces in Euclidean space $\mathbb{E}^{6}$ having constant scalar curvature. We prove that su…
We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.
A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermo…
We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness o…
The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study of the mean curvature flow of submanifolds of higher codimension has only recently started to receive attention. The mean curvature flow of submanifolds is the main object of investigation in this thesis, and indeed, the central results we obtain can be considered as high codimen…
A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings.
It is proved some results about existence and non existence of unit normal sections of submanifolds of the Euclidean space and sphere which associated Gauss maps are harmonic. Some applications to CMC hypersurfaces of the sphere and isoparametric submanifolds are obtained too.
In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y. L. Ou and L. Tang in \cite{Ou-Ta}. However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: 1. Any complete biharmonic submanifold (resp. hypersurface) $(M, g)$ in a Riemannian manifold $(N, h)$ with non-positive sectional curvature (resp. Ricci curvature) wh…
Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurface. In this paper we define $H$-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector $H$. A characterization, used by B.Y. Chen [2], [3] as the definition of strictly $H$-convexity, it is obtai…
We obtain explicit solutions of the mean curvature flow in some submanifolds of the Euclidean space. We give particularly an explicit solution of the flow of a hypersurface in the Lagrangian self-expander $L$ which is constructed in the article of Joyce, Lee and Tsui and show that it converge to a minimal one.
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is intr…
