New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
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.
Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean spa…
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently smal…
Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.
problem Preserving positive sectional curvature in contracting curvature flows in hyperbolic space.
method Homogeneous speed flow with positive sectional curvature, including kth mean curvature flow. result Positive sectional curvature is preserved and the hypersurface contracts to a round point in finite time.
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
The paper proves stabilization in hypersurface sections using Grothendieck rings and probabilistic methods.
problem Stabilization of configuration spaces and Hodge Euler characteristics for hypersurface sections.
method Geometric and cohomological stabilization, probabilistic interpretation, Grothendieck ring analysis.
result Explicit formulas for stable values of Hodge Euler characteristics and point counts.
Minimal hypersurfaces are the only H-tensional in 4D space forms.
problem Classifying H-tensional hypersurfaces in 4D space forms. method Investigation of H-tensional hypersurfaces in 4-dimensional space forms of constant sectional curvature. result Minimal hypersurfaces are the only H-tensional hypersurfaces in 4D space forms. Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.
problem Characterizing hypersurfaces in pseudo-Euclidean space.
method Defined and studied warped product hypersurfaces with constant sectional curvature or rotational properties.
result Hypersurfaces in pseudo-Euclidean space either have constant curvature or are contained in rotational hypersurfaces.
We study Kaehlerian manifolds with Norden metric g and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of (R2n+2,g,J) with constant totally real sectional curvatures.
Identifies null hypersurfaces with constant surface gravity.
problem Understanding null hypersurfaces in spacetimes.
method Analyzes spacetimes satisfying null convergence condition.
result Null hypersurfaces admit null sections with constant surface gravity.
We classify the hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$ with constant sectional curvature and dimension n≥3.
The paper classifies hypersurfaces in H2imesH2 with constant curvature.
problem Classifying hypersurfaces in H2imesH2 with constant sectional curvature. method Analyzing the geometry of H2imesH2 and constructing specific examples. result Examples of hypersurfaces in H2imesH2 with non-constant product angle function. Sharp lower bound for curvature in Kähler manifolds.
problem Curvature bounds for real hypersurfaces in Kähler manifolds.
method Gauss equation for semi-isometric CR immersions.
result Proves $ rac12$-positivity of Tanaka-Webster scalar curvature.
Study on the geometry of spacelike hypersurfaces in spacetime.
problem Understanding the geometry of compact spacelike Cauchy hypersurfaces.
method Analysis of a weak Riemannian metric on the manifold of hypersurfaces.
result Positive geodesic distance and non-positive sectional curvature.
The paper classifies hypersurfaces in a product of two spheres with constant curvature.
problem Classifying hypersurfaces in S2imesS2 with constant sectional curvature. method Applying the Tsinghua principle and solving the sinh-Gordon equation.
result Hypersurfaces with constant sectional curvature are parallel to minimal hypersurfaces with C=0. New proof of Minkowski identities for hypersurfaces in curved spaces.
problem Proving Minkowski identities for hypersurfaces in constant curvature manifolds.
method Using a differential system and position vector field.
result New proof of Minkowski identities for hypersurfaces.
New method bounds sectional curvatures and proves Spectral Theorem.
problem Computing and bounding sectional curvatures for algebraic curvature tensors.
method Algebraic and geometric methods to construct hypersurfaces with prescribed sectional curvatures.
result A relatively short proof of the Spectral Theorem for self-adjoint operators.
Study of null hypersurfaces in semi-Riemannian manifolds with applications.
problem Characterize null hypersurfaces in semi-Riemannian manifolds with varying sectional curvature.
method Introduce screen quasi-conformal hypersurfaces and extend results to non-zero sectional curvature.
result Classification results for umbilical, isoparametric, and Einstein null hypersurfaces in Lorentzian space forms.
Study of trapped submanifolds in null hypersurfaces with curvature constraints.
problem Understanding trapped submanifolds in null hypersurfaces with curvature constraints.
method Analyzing null hypersurfaces with constant sectional curvature and proving properties of trapped submanifolds.
result Null trapping horizons cannot exist in null hypersurfaces with non-positive curvature.
The paper classifies Einstein hypersurfaces in two types of warped product manifolds.
problem Classifying Einstein hypersurfaces in SnimesR and HnimesR. method Characterization of hypersurfaces with principal directions and isoparametric hypersurfaces theory.
result Einstein hypersurfaces in SnimesR and HnimesR have constant sectional curvature. We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…
Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
problem Characterizing hypersurfaces with constant curvature in six-dimensional nearly Kähler manifolds.
method Proving hypersurfaces are η-quasi umbilical in specific spaces, then using non-existence results. result Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
Hypersurfaces of manifolds of constant nonzero sectional curvature are classificated according their restricted homogeneous holonomy groups.
We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…
Study on real hypersurfaces in complex projective plane with constant mean curvature.
problem Real hypersurfaces in complex projective plane satisfying a specific inequality involving δ(2).
method Analyzing non-Hopf real hypersurfaces with constant mean curvature along Reeb vector field integral curves.
result Description of all such hypersurfaces satisfying the equality case.
Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sec…
We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus pro…
Study proves rigidity of certain hypersurfaces in 5- and 6-manifolds.
problem Proving rigidity of stable minimal hypersurfaces in 5- and 6-manifolds.
method Nonnegative 3-intermediate Ricci curvature combined with uniformly positive k-triRic curvature.
result No complete noncompact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature.
The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.
problem Constructing and studying the Gromoll-Meyer sphere in a specific homogeneous space.
method Constructing a homogeneous isoparametric foliation and transnormal system on the quotient space, analyzing the induced metrics.
result The Gromoll-Meyer sphere has positive Ricci curvature and quasi-positive sectional curvature.
The study explores Einstein hypersurfaces in warped product spaces and their properties.
problem Investigating Einstein hypersurfaces in warped product spaces.
method Analyzing constant curvature hypersurfaces and properties of gradient functions.
result Characterization of ideal Einstein hypersurfaces with specific curvature properties.
Revisits and applies a formula for hypersurface Euler characteristics.
problem Understanding isoparametric hypersurfaces in space forms.
method Re-proves Allendoerfer-Weil's formula and applies it to isoparametric hypersurfaces.
result New insights into isoparametric hypersurfaces.
New complete hypersurfaces found in affine geometry.
problem Finding new complete hypersurfaces in affine geometry.
method Classifying special Calabi hypersurfaces and solving equations.
result Found a class of new Euclidean and Calabi complete affine hypersurfaces.
In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces w…
The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.
problem Characterizing properties of minimal hypersurfaces in higher-dimensional Riemannian manifolds.
method Maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces.
result Two disjoint properly embedded minimal hypersurfaces bound a slab in specific conditions.
The paper studies Einstein hypersurfaces in a specific warped product space.
problem Investigating Einstein hypersurfaces in a warped product space.
method Analyzing the principal curvatures and using multiply warped product structure.
result Hypersurfaces have at most three distinct principal curvatures and are locally multiply warped products.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In th…
Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Totally geodesic hypersurfaces in hyperbolic manifolds are rigid under certain conditions.
problem Conditions under which totally geodesic hypersurfaces in hyperbolic manifolds are rigid.
method Study of homotopy equivalence and sectional curvature properties.
result Conditions for rigidity of totally geodesic hypersurfaces in hyperbolic manifolds.
In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a C2-smooth compact hypersurface S with - or without - boundary ∂S; S and ∂S are endowed with their homo…
Study of f-biharmonic hypersurfaces in conformally flat spaces.
problem Characterize f-biharmonic hypersurfaces in conformally flat spaces. method Analyze f-biharmonicity of totally umbilical hypersurfaces in various contexts. result Properties of f-biharmonic hypersurfaces in nonpositively curved manifolds. Study on harmonic Gauss maps for submanifolds in Euclidean space and sphere.
problem Existence and non-existence of unit normal sections with harmonic associated Gauss maps.
method Proof of existence and non-existence results for submanifolds in Euclidean space and sphere.
result Obtained applications to CMC hypersurfaces of the sphere and isoparametric submanifolds.
New types of null hypersurfaces found in Sasakian space-forms.
problem Identifying new structures in Sasakian space-forms.
method Defined and analyzed contact screen conformal and umbilic null hypersurfaces.
result Proved these hypersurfaces exist in Sasakian space forms with specific curvature.
Paper proves new rigidity results for biconservative hypersurfaces.
problem Rigidity of non-negatively curved compact biconservative hypersurfaces.
method Alternative proofs and new estimates of the squared norm of the shape operator.
result New rigidity results for biconservative hypersurfaces in space forms.
Classifies hypersurfaces with constant curvature in product spaces.
problem Classifying hypersurfaces with constant curvature in product spaces.
method Analyzing hypersurfaces in R^k x S^{n-k+1} and R^k x H^{n-k+1} for 2 <= k <= n-1.
result Complete description of hypersurfaces with constant curvature in product spaces of space forms.
The paper finds conditions for certain hypersurfaces to be totally umbilical.
problem Conditions for constant mean curvature hypersurfaces to be totally umbilical.
method Analyzes the traceless part of the second fundamental form.
result Establishes conditions for complete constant mean curvature hypersurfaces to be totally umbilical.
The paper studies critical sections of the Allen-Cahn functional and their relation to minimal hypersurfaces.
problem Minimal hypersurfaces with boundary equal to a given submanifold.
method Analysis of the Allen-Cahn functional and its critical sections.
result The limit of critical sections converges to a stationary varifold, which is a minimal hypersurface away from the boundary.