Constructs new explicit proper r-harmonic functions on Thurston geometries.
problem Developing proper r-harmonic functions on specific 3D geometries.
method Explicit construction of new functions for Thurston geometries.
result Explicit construction of new proper r-harmonic functions on various Thurston geometries.
Polyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i:Sn−1(R)→Sn i…
New method constructs complex-valued r-harmonic functions on Riemannian manifolds.
problem Constructing complex-valued r-harmonic functions on Riemannian manifolds.
method Introducing a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds and applying it to specific semisimple Lie groups.
result The method successfully constructs complex-valued r-harmonic functions on various Riemannian manifolds, including specific Lie groups.
The paper studies polyharmonic hypersurfaces in space forms, proving their minimal properties and characterizing specific cases.
problem Characterizing and understanding polyharmonic hypersurfaces in space forms.
method Analyzing hypersurfaces of order r (briefly, r-harmonic) in space forms Nm+1(c), focusing on c≤0 and Sm+1. result Proves that r-harmonic hypersurfaces in Nm+1(c) are minimal if c≤0 and mean curvature and shape operator are constant. Constructs biharmonic and r-harmonic submanifolds in cohomogeneity one manifolds.
problem Constructing biharmonic and r-harmonic submanifolds. method Using cohomogeneity one manifolds, the normal index of submanifolds is studied, and new examples are provided.
result Constructs metrics on the sphere with biharmonic non-minimal hypersurfaces.
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. The paper explores polyharmonic hypersurfaces in pseudo-Riemannian space forms.
problem Characterizing polyharmonic hypersurfaces in pseudo-Riemannian space forms.
method Analyzing hypersurfaces with specific properties under given conditions.
result Existence of new families of proper r-harmonic hypersurfaces.
The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.
problem Existence and classification of polyharmonic helices of order r.
method Analytical and geometric approaches, including Lie group theory and Euclidean sphere analysis.
result Complete classification of proper r-harmonic helices in Sol_3 and new examples in Bianchi-Cartan-Vranceanu spaces.
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 construct new explicit proper r-harmonic functions on the standard n-dimensional sphere S^n and hyperbolic space H^n for any r\ge 1 and n\ge 2.
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 explores polyharmonic curves in semi-Riemannian manifolds.
problem Investigating polyharmonic curves in semi-Riemannian manifolds.
method Analyzing Frenet curves in semi-Riemannian manifolds of various types.
result Existence, non-existence, and classification results for polyharmonic curves.
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
The study describes the geometry of surfaces and their representations in SL(3,R).
problem Understanding the geometry of surface group representations into SL(3,R).
method Proving asymptotic formulas and harmonic map convergence for equivariant maps.
result The geometry of the image is weakly convex and a (one-third) translation surface.
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane R2 with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…
The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called ES−r-energy funct…