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.
This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, tho…
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…
The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.
problem Proving that biharmonic functions with certain growth conditions are constant or harmonic.
method Using a new local L2 estimate for the Laplacian of biharmonic functions combined with a mean value inequality. result Any biharmonic function of subquadratic growth on a manifold with nonnegative Ricci curvature must be harmonic, and any of sublinear growth must be constant.
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
The paper studies biharmonic functions and bi-eigenfunctions on spheres and model spaces.
problem Characterizing biharmonic functions and eigenfunctions on model spaces.
method Analyzes bi-Laplacian on spheres, derives integral formulas for biharmonic solutions, and classifies proper biharmonic functions.
result Proper biharmonic functions on model spaces can be constructed from eigenfunctions of the factor sphere.
Finite-dimensional spaces of biharmonic functions on manifolds are explored.
problem Characterizing biharmonic functions on open manifolds with nonnegative Ricci curvature.
method Analyzing bounded and polynomial growth biharmonic functions, deriving Weyl bounds, and studying fourth-order operators.
result Finite-dimensional spaces of biharmonic functions with polynomial growth are established.
Proves generalized Chen's conjecture for biharmonic maps on foliations.
problem Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
method Analyzes (F,F')-biharmonic maps and their critical points.
result Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
We construct new proper biharmonic functions defined on open and dense subsets of the special unitary group SU(2). Then we employ a duality principle to obtain new proper biharmonic functions from the non-compact 3-dimensional hyperbolic space H^3.
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 …
We construct a new class of biharmonic maps, which are the critical points for the bienergy functional, by deforming conformally the codomain metric of harmonic Riemannian submersions such that they become nonharmonic but biharmonic.
The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form …
Study biharmonic submanifolds in warped product structures.
problem Characterize biharmonic submanifolds in warped product spaces.
method Analyze tension and bitension fields, relate to warping function and geometry of submanifolds.
result Characterize tangentially and normally biharmonic cases via differential conditions on the warping function.
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.
Author presents the second variational formula for statistical biharmonic maps.
problem Developing a formula for statistical biharmonic maps.
method Introduced the second variational formula for the statistical bi-energy functional.
result The second variational formula can be represented using Hessian curvature in Hessian manifolds.
We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bu…
The paper studies conformal-biharmonic hypersurfaces in spheres and product spaces.
problem Characterizing conformal-biharmonic hypersurfaces in spheres and product spaces.
method Analyzing critical points of the conformal-bienergy functional and studying properties of hypersurfaces in product spaces.
result Characterization of conformal-biharmonic hypersurfaces in spheres and product spaces.
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.
We construct new explicit proper biharmonic functions on the 3-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.
The notions of bienergy of a smooth mapping and of biharmonic map between Riemannian manifolds are extended to the case when the domain is Finslerian. We determine the first and the second variation of the bienergy functional, the equations of Finsler-to-Riemann biharmonic maps and some specific examples. Two notable r…
The paper characterizes biharmonic maps between spheres using polynomial functions.
problem Characterizing biharmonic maps between spheres using polynomial functions.
method Proved a characterization formula and constructed biharmonic maps.
result Classification of all proper biharmonic quadratic forms from spheres.
In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of \cite{OW} and \cite{FOR}. We derived the biharmonic equation for hy…
Let Ω be a bounded domain with C∞ boundary in an n-dimensional C∞ Riemannian manifold, and let ϱ be a non-negative bounded function defined on ∂Ω. It is well-known that for the biharmonic equation Δ2u=0 in Ω with the 0-Dirichlet boundary condition, there exists an infinite se…
The paper characterizes biharmonic submersions from product manifolds.
problem Characterizing biharmonic Riemannian submersions from M2imesR. method Local characterizations and by-products of biharmonic submersions.
result Local characterizations of biharmonic Riemannian submersions and uniqueness of a specific submersion.
Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese…
Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $φ:\s^3\to \s^3$. We show φ to be unstable and estimate its biharmonic index and nullity. Resolving the s…
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. Study on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.
problem Investigating higher-order harmonicity in pseudo-Riemannian geometry.
method Deriving first variations of bienergy and interpolating sesqui-energy functionals, characterizing biharmonic and interpolating sesqui-harmonic vector fields.
result Explicit characterizations and examples of vector fields satisfying biharmonic and interpolating sesqui-harmonic conditions.
Stability of biharmonic maps in critical dimension proven.
problem Stability of biharmonic maps between manifolds in critical dimension.
method Generalization of Morse stability theory to biharmonic maps, development of strong energy quantization method.
result Strong energy quantization in a wide class of problems in geometric analysis.
Study of loops in sums of Laplace eigenfunctions on surfaces.
problem Uniform bound for the number of nested loops in sums of Laplace eigenfunctions.
method Real-analytic category analysis and biharmonic function construction.
result Uniform bound for the number of rooted double nests in terms of surface, root, and spectral cutoff.
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using …
We study biharmonic maps and f-biharmonic maps from a round sphere (S2,g0), the latter maps are equivalent to biharmonic maps from Riemann spheres (S2,f−1g0). 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-biharmonic maps and submersions in space forms.
problem Characterizing f-biharmonic maps and submersions in space forms. method Analyzing f-biharmonic curves, developing classifications, and using integrability data. result Proper f-biharmonic developable surfaces exist only in the case of cylinders. Study of p-biharmonic curves and their properties.
problem Generalizing biharmonic curves to p-biharmonic curves. method Classification and analysis of p-biharmonic curves on surfaces and space forms. result Existence and stability of p-biharmonic curves on closed surfaces. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an n-dimensional Riemannian manifold.
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-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 fourth order Lamm-Riviere system for biharmonic mappings in 4D.
problem Higher order regularity and sharp Holder continuity of weak solutions.
method Optimal higher order regularity and sharp Holder continuity through analysis of the Lamm-Riviere system.
result Derive weak compactness for sequences of weak solutions with uniformly bounded energy.
We consider the energy and bienergy functionals as variational problems on the set of Riemannian metrics and present a study of the biharmonic stress-energy tensor. This approach is then applied to characterise weak conformality of the Gauss map of a submanifold. Finally, working at the level of functionals, we recover…
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.
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 identity map of certain Einstein manifolds is stable in both energy and bienergy.
problem Stability of the identity map in Einstein manifolds.
method Investigation of conformal-biharmonic stability compared to harmonic stability.
result The conformal-biharmonic index coincides with the harmonic index, except for the 4D Euclidean sphere.
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.