The study examines stability of triharmonic hypersurfaces in space forms.
problem Stability of triharmonic hypersurfaces in space forms.
method Derivation of general stability statements, focus on specific cases of constant mean curvature in Euclidean and hyperbolic spaces, and analysis of small proper triharmonic hyperspheres and Clifford tori.
result Triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations, while in hyperbolic space they are stable.
Study on triharmonic curves in f-Kenmotsu manifolds.
problem Characterizing triharmonic curves in f-Kenmotsu manifolds.
method Investigation of necessary and sufficient conditions for Frenet curves, slant, and Legendre curves to be triharmonic. Proof of specific properties of triharmonic Frenet curves.
result Triharmonic Frenet curves with constant curvature are Frenet helices in three dimensional f-Kenmotsu manifolds.
Study on triharmonic curves in 3D spaces, proving their existence and classification.
problem Characterizing triharmonic curves in 3D homogeneous spaces.
method Analyzing curves with constant curvature in Riemannian manifolds, focusing on Frenet helices and space forms.
result Classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.
Study on triharmonic curves in Sol space with constant curvature and torsion.
problem Characterizing triharmonic curves in the Sol space.
method Complete classification of proper triharmonic curves with constant geodesic curvature and torsion.
result Triharmonic curves form a constant angle with a Killing field of constant length.
The paper studies triharmonic hypersurfaces in space forms and proves their properties.
problem Characterizing triharmonic hypersurfaces in different space forms.
method Analyzing hypersurfaces in spheres, hyperbolic spaces, and Euclidean spaces using CMC (constant mean curvature) and triharmonic properties.
result Properties of triharmonic hypersurfaces in Euclidean space, including minimality.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.
problem Characterizing triharmonic CMC hypersurfaces with distinct principal curvatures.
method Analyzing critical points of the tri-energy and applying geometric properties.
result Proves conditions for constant scalar curvature and minimality of hypersurfaces.
Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
We derive a monotonicity formula and classify finite Morse index solutions (positive or sign-changing, radial or not) to the following triharmonic Lane-Emden equation: \begin{equation}\nonumber (-Δ)^3 u=|u|^{p-1}u \hbox{ in } \mathbb{R}^n, \end{equation} where p is below the Joseph-Lundgren exponent. As a byproduct w…
A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are fin…
The paper proves properties of triharmonic CMC hypersurfaces with limited curvature types.
problem Characterizing triharmonic CMC hypersurfaces with specific curvature constraints.
method Analyzing critical points of the triharmonic energy and applying geometric inequalities.
result Proves constant scalar curvature for triharmonic CMC hypersurfaces with at most 3 distinct principal curvatures.
For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemanni…
We derive the stress-energy tensor for polyharmonic maps between Riemannian manifolds. Moreover, we employ the stress-energy tensor to characterize polyharmonic maps where we pay special attention to triharmonic maps.
Study classifies polyharmonic helices in various space forms.
problem Classifying polyharmonic helices in different space forms.
method Derived classification results for polyharmonic helices in space forms.
result Polyharmonic helices of arbitrary order in space forms of negative curvature are geodesics.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
The paper proves existence and instability of weak r-harmonic maps.
problem Existence and stability of weak r-harmonic maps. method Construction of critical points and analysis of stability.
result Existence and instability of weak r-harmonic maps restricted to specific dimensions.