The paper finds conditions for biharmonic orbits in symmetric spaces.
problem Conditions for biharmonic orbits in symmetric spaces.
method Analyzes isotropy representations and biharmonic submanifolds in hyperspheres.
result Necessary and sufficient conditions for biharmonic orbits in symmetric spaces.
The study identifies biharmonic submanifolds in compact symmetric spaces and Lie groups.
problem Characterizing biharmonic submanifolds in compact symmetric spaces and Lie groups.
method Using necessary and sufficient conditions derived from symmetric triad with multiplicities.
result Determined all biharmonic submanifolds in irreducible symmetric spaces of compact type and biharmonic hypersurfaces in compact Lie groups.
In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.
Study on G 2 G_{2} G 2 actions on symmetric spaces, focusing on orbit properties.
problem Investigating properties of orbits in symmetric spaces related to G 2 G_{2} G 2 . method Classification and analysis of orbits as Riemannian submanifolds, focusing on principal curvatures and specific types of orbits.
result Classification and properties of orbits in symmetric spaces related to G 2 G_{2} G 2 . Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention and a cohomogeneity-one analytic system, proving local existence for nonconstant mean curvature and dilation.
result Local existence and rigidity results for biharmonic conformal immersions into anti-de Sitter space.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention, expressing biharmonic equation in terms of induced metric and curvature, deriving cohomogeneity-one analytic system, and solving scalar third-order ODE.
result Local existence and rigidity of biharmonic conformal immersions with nonconstant dilation.
The aim of this paper is to prove that there exists no cohomogeneity one G − G- G − invariant proper biharmonic hypersurface into the Euclidean space R n {\mathbb R}^n R n , where G G G denotes a tranformation group which acts on R n {\mathbb R}^n R n by isometries, with codimension two principal orbits. This result may be considered in the…
Study f-biharmonic maps and submanifolds, proving rigidity and classifications.
problem Characterize and classify f-biharmonic maps and submanifolds.
method Explore applications, use improved equations, and analyze conformal changes.
result Prove rigidity theorems and classifications of f-biharmonic hypersurfaces.
f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic maps. In this paper, we prove that an f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density…
We study biharmonic maps and f-biharmonic maps from a round sphere ( S 2 , g 0 ) (S^2, g_0) ( S 2 , g 0 ) , the latter maps are equivalent to biharmonic maps from Riemann spheres ( S 2 , f − 1 g 0 ) (S^2, f^{-1}g_0) ( S 2 , f − 1 g 0 ) . We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order l…
The article explores constructing biharmonic and conformal biharmonic maps to spheres.
problem Constructing biharmonic and conformal biharmonic maps to spheres.
method Geometric algorithm to render harmonic maps biharmonic or conformally biharmonic.
result Explicit critical points for conformal-biharmonic maps between spheres are found.
Survey on biharmonic Riemannian submersions, a dual concept of biharmonic submanifolds.
problem Understanding biharmonic Riemannian submersions as a dual concept.
method Short survey focusing on biharmonic Riemannian submersions.
result Survey provides insights into the dual concept of biharmonic submanifolds.
The paper studies f f f -biharmonic maps and submersions in space forms.
problem Characterizing f f f -biharmonic maps and submersions in space forms. method Analyzing f f f -biharmonic curves, developing classifications, and using integrability data. result Proper f f f -biharmonic developable surfaces exist only in the case of cylinders. Study of p p p -biharmonic curves and their properties.
problem Generalizing biharmonic curves to p p p -biharmonic curves. method Classification and analysis of p p p -biharmonic curves on surfaces and space forms. result Existence and stability of p p p -biharmonic curves on closed surfaces. The paper classifies biharmonic immersions and submersions in specific spheres.
problem Classifying biharmonic immersions and submersions in specific spheres.
method Analyzing biharmonic isometric immersions and Riemannian submersions from Berger 3-spheres.
result Complete classification of proper biharmonic Hopf tori in Berger 3-sphere.
Study of f f f -biharmonic hypersurfaces in conformally flat spaces.
problem Characterize f f f -biharmonic hypersurfaces in conformally flat spaces. method Analyze f f f -biharmonicity of totally umbilical hypersurfaces in various contexts. result Properties of f f f -biharmonic hypersurfaces in nonpositively curved manifolds. Reviews recent work on biharmonic conformal maps and their properties.
problem Understanding biharmonic conformal maps and their properties.
method Analyzes recent research on biharmonic conformal maps, immersions, and maps between manifolds.
result Links biharmonic conformal maps to isoparametric functions and Yamabe type equations.
The paper studies biharmonic submanifolds in pseudo-Riemannian manifolds.
problem Characterizing biharmonic submanifolds in pseudo-Riemannian geometry.
method Derived biharmonic equations and proved properties of biharmonic submanifolds.
result Found constant mean curvature for pseudo-umbilical biharmonic submanifolds.
Survey on recent biharmonic submanifolds progress.
problem Chen's biharmonic conjecture and related submanifolds.
method Review of recent developments and open problems.
result Brief overview of recent progress and open challenges.
The paper classifies biharmonic hypersurfaces with constant scalar curvature.
problem Classifying biharmonic hypersurfaces with constant scalar curvature.
method Analyzing biharmonic hypersurfaces in space forms and spheres.
result Supports conjectures on biharmonic submanifolds and hypersurfaces.
Totally biharmonic hypersurfaces in space forms and 3D BCV spaces classified.
problem Characterizing totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
method Analyzing geodesics and isoparametric properties to classify hypersurfaces.
result Classification of totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
Abstract: Characterizations for submersions to be harmonic or biharmonic shown; examples provided.
problem Characterizing harmonic and biharmonic Riemannian submersions.
method Characterizations and examples provided.
result Characterizations and examples of submersions provided.
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 subma…
The study of conformal biharmonic maps and hypersurfaces in various spaces.
problem Understanding the properties and behavior of conformal biharmonic maps and hypersurfaces.
method Investigation of the conformal bienergy functional and its critical points, focusing on hypersurfaces in spheres and hyperbolic spaces.
result Identification and classification of conformal biharmonic hypersurfaces in spheres and hyperbolic spaces, including stability analysis.
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
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.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
The paper studies biharmonic submersions in Riemannian geometry.
problem Characterizing biharmonic Riemannian submersions.
method Deriving bitension field equations and using them to construct biharmonic examples and characterize submersions.
result Examples of proper biharmonic Riemannian submersions with 1-dimensional fibers and characterizations of warped products.
Extends p p p -biharmonic and bi- p p p -harmonic map definitions.
problem No specific problem stated; extends definitions.
method Extends definitions of p p p -biharmonic and bi- p p p -harmonic maps. result Properties of extended maps explored.
Proves Chen's conjecture on biharmonic submanifolds in Euclidean space and space forms.
problem Chen's conjecture on biharmonic submanifolds in Euclidean space.
method Derived a fundamental identity involving the mean curvature vector field and used it to prove the conjecture.
result Proved Chen's conjecture on biharmonic submanifolds in a Euclidean space and space forms.
The study proves unique continuation for biharmonic maps.
problem Unique continuation of biharmonic maps between manifolds.
method Proof of unique continuation results.
result Proves several unique continuation results for biharmonic maps.
We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications …
The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.
problem Characterizing biharmonic unit vector fields and sections on 2-tori.
method Analyzing variational problems for unit vector fields under conformal metrics, proving properties through homotopy classes.
result Biharmonic unit sections on 2-tori are always harmonic and exist in each homotopy class.
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter …
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
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.
No proper biharmonic CMC compact hypersurface in a specific warped product space.
problem Existence of proper biharmonic CMC hypersurfaces in a warped product space.
method Finding necessary and sufficient conditions for proper biharmonic CMC hypersurfaces in a special warped product space.
result No proper biharmonic CMC compact hypersurface exists in the specified space.
Study on biharmonic map heat flow with monotonicity formula.
problem Properties of biharmonic heat kernel and extrinsic biharmonic map heat flow.
method Derived an entropy type quantity exhibiting monotonicity behaviors.
result Monotonicity formula for extrinsic biharmonic map heat flow.
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 biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
A submanifold M M M of a Euclidean m m m -space is said to be biharmonic if Δ H → = 0 Δ\overrightarrow H=0 Δ H = 0 holds identically, where H → \overrightarrow H H is the mean curvature vector field and Δ Δ Δ is the Laplacian on M M M . In 1991, the author conjectured that every biharmonic submanifold of a Euclidean space is minimal. The study of b…
The study classifies biharmonic real hypersurfaces in complex projective spaces.
problem Classifying biharmonic real hypersurfaces in complex projective spaces.
method Analyzing proper biharmonic Hopf and ruled real hypersurfaces with distinct principal curvatures.
result Biharmonic ruled real hypersurfaces in complex projective spaces are minimal.
The study classifies equivariant biharmonic maps and proves stability results for certain maps.
problem Classifying and analyzing equivariant biharmonic maps and their stability.
method Generalized biharmonic equation for equivariant maps, improved second variation formula for biharmonic maps.
result No stable proper biharmonic maps with constant square norm of tension field exist from a compact Riemannian manifold into a space form of positive sectional curvature.
Biharmonic curves are a generalization of geodesics, with applications in elasticity theory and various branches of computer science. The paper proposes a first study of biharmonic curves in spaces with Finslerian geometry, covering the following topics: a deduction of their equations, existence of non-geodesic biharmo…
The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.
problem Eigenvalue inequalities for Laplacian and biharmonic operators on submanifolds.
method Using Sobolev inequalities to establish new eigenvalue inequalities.
result Established new inequalities for Laplacian and biharmonic eigenvalues.
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 study examines biharmonic hypersurfaces in Sasakian space forms.
problem Characterizing biharmonic hypersurfaces in Sasakian space forms.
method Analyzing biharmonic hypersurfaces with the induced metric of tensor Ricci.
result Existence conditions and properties of biharmonic hypersurfaces.