Low-degree hyperbolic hypersurfaces in complex projective space constructed.
problem Constructing hyperbolic hypersurfaces of low degree in complex projective space.
method Families of hyperbolic hypersurfaces Xd constructed with degree d. result Families of hyperbolic hypersurfaces Xd of degree d≥(2n+3)2 constructed in Pn+1(C). The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface a…
Researchers create special fibrations on Calabi-Yau hypersurfaces.
problem Understanding special Lagrangian fibrations on Calabi-Yau hypersurfaces.
method Produced special Lagrangian T^n-fibrations on Calabi-Yau hypersurfaces in the Fermat family.
result Demonstrated fibrations on generic regions of Calabi-Yau hypersurfaces in the large complex structure limit.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
We construct quasi-Einstein metrics on some hypersurface families. The hypersurfaces are circle bundles over the product of Fano, Kähler-Einstein manifolds. The quasi-Einstein metrics are related to various gradient Kähler-Ricci solitons constructed by Dancer and Wang and some Hermitian, non-Kähler, Einstein metrics co…
Minimal surfaces found in finite volume manifolds.
problem Existence of minimal hypersurfaces in complete manifolds of finite volume.
method Independent result on volume sweeping by hypersurfaces; main tool is a volume constraint result.
result Proves existence of minimal hypersurfaces in complete non-compact manifolds of finite volume.
Proves rigidity in product spaces using index theory.
problem Scalar curvature rigidity in product spaces.
method Fredholm family index theorem.
result Recover corresponding results of Clifford-linear index theory.
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…
In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. …
Study of families of lines on spheres and their focal sets.
problem Characterizing families of lines on spheres and their geometric properties.
method Analyzing submanifolds of TSn and their focal sets, using symplectic structures and sectional curvatures. result Derivation of formulas relating sectional curvatures of focal sets to differences in radii of curvature of generating hypersurfaces.
Complete classification of homogeneous real hypersurfaces in complex 3-space.
problem Classifying locally homogeneous real hypersurfaces in C3. method Classification of abstract 5-dimensional real Lie algebras and their representations by algebras of holomorphic vector fields in complex 3-space.
result 47 types of homogeneous hypersurfaces, including 1- or 2-parametric families and single hypersurfaces/families.
Constructs families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.
problem Constructing compact monotone Lagrangians in Brieskorn-Pham hypersurfaces.
method Inspired by monodromy considerations, techniques for controlling homology, Maslov class, and monotonicity constant.
result Infinite families of monotone Lagrangian S1imesΣg in C3 for g≥2. Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.
This paper constructs GCM hypersurfaces in Kerr spacetimes.
problem Extending the Kerr family stability proof to full stability.
method Concatenating a 1-parameter family of GCM spheres by solving an ODE system.
result Removes symmetry restrictions in GCM procedure.
We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's con…
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 curvatu…
The study defines and constructs hypersurfaces in a product of two space forms.
problem Characterizing hypersurfaces in a product of two space forms.
method Explicit construction using parallel families of hypersurfaces and isoparametric hypersurfaces.
result Classification of hypersurfaces with constant mean curvature and constant product angle function.
The paper classifies rotational hypersurfaces in n-space using a modified Laplacian operator.
problem Classifying rotational hypersurfaces in n-dimensional Euclidean space.
method Investigating the Gauss map of rotational hypersurfaces with respect to the operator Ln−3. result Established a classification theorem connecting the matrix A and the Gauss map G through the equation Ln−3G=AG. The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hyp…
Study shows how certain hypersurfaces evolve under mean curvature flow.
problem Evolution of isoparametric hypersurfaces under mean curvature flow.
method Reparametrization of the parallel family in short time.
result Evolution given by a reparametrization of the parallel family.
Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. 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 m…
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 spa…
The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.
problem Exploring the space of CMC hypersurfaces in spheres.
method Description and verification of CMC hypersurfaces, focusing on H=0 cases. result Verification of Yau's conjecture for minimal hypersurfaces in spheres.
Characterizes concircular helices in space forms and ruled hypersurfaces.
problem Understanding concircular hypersurfaces and helices in space forms.
method Characterization through differential equations and ruled hypersurfaces.
result Concircular helices are geodesics of concircular surfaces.
Ancient mean curvature flows start from unstable minimal hypersurfaces.
problem Constructing ancient solutions to mean curvature flow.
method From an unstable minimal hypersurface with finite total curvature in \(\mathbb{R}^{n+1}\), we construct \(I\)-dimensional families of embedded ancient solutions.
result Ancient solutions arise from unstable minimal hypersurfaces.
The paper classifies hypersurfaces with constant curvature in Euclidean spaces.
problem Classifying separable hypersurfaces with constant sectional curvature.
method Analytical proof and classification of hypersurfaces in Euclidean spaces.
result Hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
Classifies hypersurfaces in the product of two spheres.
problem Classifying hypersurfaces in S2imesS2. method Homogeneous and isoparametric hypersurfaces classification.
result A family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature.
It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. Study finds minimal hypersurface in convex manifolds with non-negative Ricci curvature.
problem Finding minimal hypersurfaces in manifolds with specific curvature properties.
method Min-max method applied to one-parameter families of hypersurfaces.
result The min-max minimal hypersurface is orientable, of index one and multiplicity one.
New formulas compare total mean curvatures of nested hypersurfaces.
problem Computing total mean curvatures of nested hypersurfaces.
method Developed differential forms based on Chern's work to compare curvatures.
result Quicker proof of recent result on total mean curvatures.
It is known that hypersurfaces in CPn or CHn for which the number g of distinct principal curvatures satisfied g≤2 must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that n≥3. In this paper, we construct a 2-parameter family of non-Hopf hypersurfaces in…
New tensors capture intrinsic embedding data of conformal hypersurfaces.
problem Classifying hypersurface invariants in conformal manifolds.
method Constructing curvatures and conformal fundamental forms.
result Finite family of tensors captures extrinsic embedding data.
In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a…
In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparam…
Geodesic flows between hypersurfaces in Euclidean spaces using Lorentzian geometry.
problem Interpolation between hypersurfaces in Euclidean spaces.
method Lorentzian geodesic flow between tangent spaces of hypersurfaces.
result Geodesic flow is preserved by rigid transformations and homotheties.
New examples of hypersurfaces found in quaternionic hyperbolic spaces.
problem Classifying actions on symmetric spaces of rank one.
method Cohomogeneity one actions and orbit equivalence.
result Uncountably many inhomogeneous isoparametric families of hypersurfaces.
Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.
problem Creating smooth CMC hypersurfaces from piecewise-smooth unions of spheres.
method Gluing totally umbilical 3-spheres to specific Clifford hypersurfaces, forming a smooth one-parameter family of CMC hypersurfaces.
result Desingularization of piecewise-smooth hypersurfaces yields smooth CMC hypersurfaces with embedded and non-embedded types.
We prove the existence of a one parameter family of minimal embedded hypersurfaces in Rn+1, for n≥3, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar end…
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
The geometry of canal hypersurfaces of an n-dimensional conformal space C^n is studied. Such hypersurfaces are envelopes of r-parameter families of hyperspheres, 1 \leq r \leq n-2. In the present paper the conditions that characterize canal hypersurfaces, and which were known earlier, are made more precise. The main at…
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…