We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.
In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.
In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that such submanifolds are isoparametric.
We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.
We investigate behavior of principal curvatures and principal vectors near a non-degenerate singular point of the first kind of frontals. As an application, we extend the notion of Ribaucour transformations to frontals with singular points.
We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.
In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: is every hypersurface with constant principal curvatures isoparametric?
In this paper, we have studied biharmonic hypersurfaces in space form $\bar{M}^{n+1}(c)$ with constant sectional curvature $c$. We have obtained that biharmonic hypersurfaces $M^{n}$ with at most three distinct principal curvatures in $\bar{M}^{n+1}(c)$ has constant mean curvature. We also obtain the full classification of biharmonic hypersurfaces with at most three distinct principal curvatures in arbitrary dimension space form $\bar{M}^{n+1}(c)$.
If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, we prove that if $M$ is an irreducible connected proper Dupin hypersurface in $\R^n$ (or $S^n$) with four distinct principal curvatures with multiplicities $m_1=m_2 \geq 1$ and $m_3=m_4=1$, and constant Lie curvature $r=-1$, then $M$ is equivalent by Lie sphere transformation to a…
We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.
We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones.
We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal plane. In some cases, a kind of trichotomy phenomenon is observed: the curves of the foliations may be circles, hyperbolas or horocycles, depending of whether the constant principal curvature is respectively larger, smaller of equal to one (not necessarily in this order). We descr…
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures. We also give explicit examples.
Let $x$ be an $m$-dimensional umbilic-free hypersurface in an $(m+1)$-dimensional unit sphere $\mathbb{S}^{m+1}(m\geq3)$. One of important questions is to classify hypersurfaces with two distinct principal curvatures. In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Moebius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.
It is known that hypersurfaces in $CP^n$ or $CH^n$ for which the number $g$ of distinct principal curvatures satisfied $g \le 2$ must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that $n \ge 3$. In this paper, we construct a 2-parameter family of non-Hopf hypersurfaces in $CP^2$ and $CH^2$ with $g=2$ and show that every non-Hopf hypersurface with $g=2$ is locally of this form.
We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces $\mathbb S^{n+1}$ are either the hypersphere $\mathbb S^n(1/\sqrt2)$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show that there does not exi…
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…
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships betwe…
Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any specified value of the Hopf principal curvature less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in CH^2.
In [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of two polynomials is a non-zero polynomial. In this paper we point out that, in fact, there is a case, and only one, when this resultant is the zero polynomial and therefore the original proof is not fully complete. Further, we prove that in this special case we still obtain that the hypersurface has constant mean curvature.
In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}\subset \mathbb{R}^{n+2}$ and $\mathbb{H}^{n}\times\mathbb{R}\subset \mathbb{L}^{n+2}$ with f…
We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal …
We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces.
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 …
The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvatures ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $δ(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $δ(2)$-ideal hypersurface i…
In this paper, we investigate Einstein hypersurfaces of the warped product $I\times_{f}\mathbb{Q}^{n}(c)$, where $\mathbb{Q}^{n}(c)$ is a space form of curvature $c$. We prove that $M$ has at most three distinct principal curvatures and that it is locally a multiply warped product with at most two fibers. We also show that exactly one or two principal curvatures on an open set imply constant sectional curvature on that set. For exactly three distinct principal curvatures this is no longer true, and we classify such hypersurfaces provided it does not have constant sectional curvature and a cert…
We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7,8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated in the late 1930s.
Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.
We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution of the Lund-Regge reduction problem for time-like W-surfaces with real principal curvatures in Minkowski space. We apply this theory to linear fractional time-like W-surfaces and obtain the natur…
We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove th…
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.
We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio $a:=κ_1/κ_2$ of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. A key ingredient in the successful determination of these surfaces is the proper choice of generating profiles. We employ the contours for parallel projection orthogonal to the helical axis. This has the advantage that the CRPC property can be nicely expressed with the help of the involution of conjugate surface tangents. The arising ordinary differential equa…
Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy $m_2 \geq 3m_1 - 1$, then the isoparametric hypersurface $M$ must be of FKM-type…
In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $κ_{1}=mκ_{2}+n$, where $m$ and $n$ are constant and $κ_{1}$ and $κ_{2}$ denote the principal curvatures at each point of the surface. We study the family of surfaces foliated by a uniparametric family of circles in parallel planes. We prove that the surface must be rotational or the surface is part of the family of Riemann examples of maximal surfaces ($m=-1$, $n=0$). Finally, we consider the class of rotational surfaces for the case $n=0$, obtaining a fir…
We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.
Given a non circular spacial closed curve whose total torsion is an integer multiple of $2π$, we construct a germ of a smooth surface that contains it as a hyperbolic principal cycle.
We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with at most two constant principal curvatures and, under certain conditions, with three constant principal curvatures.
In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectivel…
We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $σ_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $\mathbb{R}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times\mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $σ_{n-1}$ curvature and bounded principal curvatures.
The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore,…
In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-Münzner \cite{FKM}, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler, both structurally and technically, proof \cite{CCJ} that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\geq 2m_1-1$ is of OT-FKM type, which left…
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional $(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant principal Ricci curvatures is a Riemannian locally symmetric space.
Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable assumptions we prove the long time existence of the flow and the exponential convergence of the corresponding graphs in the $C^\infty$-topology to a hypersurface of constant F-curvature. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfac…
The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair $\{4,5\},\{6,9\}$ or $\{7,8\}$ in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $\{4,5\}$ in $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6,9\}$ in $S^{31}$ is either the inhomogen…
In this paper, we investigate a curvature-adapted and proper complex equifocal submanifold in a symmetric space of non-compact type. The class of these submanifolds contains principal orbits of Hermann type actions as homogeneous examples. In future, the results in this paper will be used to give a submanifold geometrical characterization of principal orbits of Hermann type actions.
In this paper, we study hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$. We first classify the hypersurfaces with constant principal curvatures and constant product angle function. Then, we classify homogeneous hypersurfaces and isoparametric hypersurfaces, respectively. Finally, we classify the hypersurfaces with at most two distinct constant principal curvatures, as well as those with three constant principal curvatures under some additional conditions.
