Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…
The study examines properties of biharmonic hypersurfaces with torse-forming vector fields.
problem Properties of biharmonic hypersurfaces in Riemannian manifolds.
method Investigates biharmonic hypersurfaces with torse-forming vector fields.
result Provides properties of biharmonic hypersurfaces with torse-forming vector fields.
Classification theorems for Ricci solitons on Minkowski hypersurfaces.
problem Characterizing Ricci solitons on pseudo-Riemannian hypersurfaces in Minkowski space.
method Analyzing the shape operator and potential vector field of hypersurfaces.
result Characterization of Ricci solitons on various types of hypersurfaces in Minkowski space.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
Lower bound found for eigenvalue of hypersurface in Riemannian manifold.
problem Finding bounds for eigenvalues of hypersurfaces in Riemannian manifolds.
method Used minimally embedded hypersurface and Ricci curvature constraints.
result Provided a lower bound for the first eigenvalue.
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.
The paper constructs all cmc hypersurfaces with two principal curvatures.
problem Finding all hypersurfaces with constant mean curvature and two principal curvatures.
method Explicit immersions and parameter analysis for hypersurfaces in space forms.
result The family of cmc hypersurfaces with two principal curvatures depends on two parameters, H and C.
The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.
problem Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
method The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres using vanishing and estimation theorems for Betti numbers.
result Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
The paper classifies and explores isoparametric hypersurfaces in pseudo-Riemannian space forms.
problem Investigating isoparametric hypersurfaces in pseudo-Riemannian space forms.
method Petrov's classification theorem and shape operator analysis.
result No isoparametric hypersurfaces of index 2 with complex principal curvatures exist.
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…
The paper studies stability and index of biharmonic hypersurfaces in Riemannian manifolds.
problem Analyzing stability and index of biharmonic hypersurfaces.
method Using a second variation formula for biharmonic hypersurfaces, computing stability index, and proving non-existence of unstable hypersurfaces.
result Proves non-existence of unstable proper biharmonic hypersurfaces in Euclidean space or hyperbolic space.
The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
problem Characterizing PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
method Analyzing the properties of hypersurfaces with at most two distinct principal curvatures.
result PMCV hypersurfaces are either minimal or locally isoparametric.
Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
In our article, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces. We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hyp…
Paper estimates curvature of minimal surfaces in a specific geometric space.
problem Estimating curvature of minimal hypersurfaces in Heisenberg groups.
method Extending Simons formula and Kato inequality to sub-Riemannian setting, applying to stable hypersurfaces.
result Integral curvature estimates for stable hypersurfaces in Heisenberg groups.
Given a null hypersurface L L L of a Lorentzian manifold, we construct a Riemannian metric g ~ \widetilde{g} g on it from a fixed transverse vector field ζ ζ ζ . We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold ( L , g ~ ) (L,\widetilde{g}) ( L , g ) and the vector field ζ ζ ζ . As an application, we prove so…
Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.
problem Estimating areas of stable hypersurfaces in curved spaces.
method Analyzes stable CMC hypersurfaces in Riemannian manifolds with specific curvature conditions.
result Derives upper bounds for the bottom spectrum of hypersurfaces.
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.
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.
The paper examines 4D hypersurfaces with constant mean curvature in pseudo-Riemannian space forms.
problem Investigating properties of 4D hypersurfaces with specific curvature conditions.
method Analyzing hypersurfaces with proper mean curvature vector field in pseudo-Riemannian space forms.
result Bi-harmonic hypersurfaces in N^5_s(c) are minimal in certain cases.
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M M 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 M M has sufficiently smal…
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-Rie…
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.
problem Finding sets of least perimeter in Riemannian manifolds.
method Short proof using de Giorgi and Miranda's tradition in flat space.
result Reduced boundary of least perimeter sets is a smooth minimal hypersurface in low dimensions.
Extends biharmonic concepts to new hypersurfaces and curves.
problem Characterize new types of hypersurfaces and curves in Riemannian manifolds.
method Extend p p p -biharmonic maps and hypersurfaces to ( p , q ) (p,q) ( p , q ) -harmonic hypersurfaces and curves, providing new examples. result New explicit examples of proper ( p , q ) (p,q) ( p , q ) -harmonic hypersurfaces and curves in space forms. 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.
Study on compact biconservative hypersurfaces in de Sitter space.
problem Understanding compact biconservative hypersurfaces in de Sitter space.
method Investigation of hypersurfaces with constant scalar curvature under geometric constraints.
result Extension of rigidity properties of biconservative hypersurfaces.
The paper explores p-biharmonic hypersurfaces in Einstein and conformally flat spaces.
problem Characterizing p-biharmonic submanifolds in Einstein spaces.
method Analyzing properties and constructing examples of p-biharmonic hypersurfaces.
result New examples of proper p-biharmonic hypersurfaces constructed.
The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.
problem Finding regularity of minimal hypersurfaces in Riemannian manifolds.
method Estimate for a one-parameter min-max minimal hypersurface.
result Generic regularity of minimal hypersurfaces in 8-dimensional Riemannian manifolds with positive Ricci curvature.
The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
The existence of closed hypersurfaces of prescribed curvature in semi-riemannian manifolds is proved provided there are barriers.
In this paper, we study biconservative hypersurfaces in S n \mathbb S^{n} S n and H n \mathbb H^{n} H n . Further, we obtain complete explicit classification of biconservative hypersurfaces in 4 4 4 -dimensional Riemannian space form with exactly three distinct principal curvatures.
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.
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 C P 3 \mathbb{C}P^3 C P 3 . result All extrinsically homogeneous hypersurfaces are classified in all homogeneous C P 3 \mathbb{C}P^3 C P 3 spaces. The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
Infinite G G G -invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G G G -homology class admits infinitely many distinct realizations by embedded minimal G G G -hypersurfaces. The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.
problem Characterizing biharmonic conformal hypersurfaces in Riemannian manifolds.
method Deriving biharmonic equations and proving properties of conformal immersions.
result Properties of biharmonic conformal hypersurfaces in space forms.
The paper proves that for a given metric, there exists another metric where the number of constant mean curvature hypersurfaces increases.
problem Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics.
method Proves the existence of a new metric such that the number of c − C M C c-CMC c − C M C hypersurfaces increases. result There exists a metric h h h such that the number of c − C M C c-CMC c − C M C hypersurfaces in ( M , h ) (M,h) ( M , h ) is strictly greater than in ( M , g ) (M,g) ( M , g ) . The paper classifies hypersurfaces in Riemannian manifolds with constant inner product and torse-forming axes.
problem Understanding hypersurfaces in Riemannian manifolds with specific geometric properties.
method Analyzing hypersurfaces with constant inner product and torse-forming axes.
result Classification of hypersurfaces with torse-forming axes.
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
The paper classifies conformal solitons in pseudo-Euclidean spaces.
problem Classifying conformal solitons in pseudo-Euclidean spaces.
method Classification through pseudo-Riemannian hypersurfaces and position vector fields.
result Complete classification of conformal solitons in pseudo-Euclidean spaces.
In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…