arXiv research

Study biharmonic functions on vector bundles with spherical symmetry.

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…

Extends Milnor's criterion to biharmonic functions.

The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

The paper studies biharmonic functions and bi-eigenfunctions on spheres and model spaces.

Finite-dimensional spaces of biharmonic functions on manifolds are explored.

Proves generalized Chen's conjecture for biharmonic maps on foliations.

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.

The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.

Author presents the second variational formula for statistical biharmonic maps.

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.

The study of conformal biharmonic maps and hypersurfaces in various spaces.

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.

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 $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial Ω$. It is well-known that for the biharmonic equation $Δ^2 u=0$ in $Ω$ with the 0-Dirichlet boundary condition, there exists an infinite se…

The paper characterizes biharmonic submersions from product manifolds.

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.

Study biharmonic vector fields and unit vector fields on Riemannian manifolds.

Study on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.

Stability of biharmonic maps in critical dimension proven.

Study of loops in sums of Laplace eigenfunctions on surfaces.

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 $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. 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.

Survey on biharmonic Riemannian submersions, a dual concept of biharmonic submanifolds.

The paper studies $f$-biharmonic maps and submersions in space forms.

Study of $p$-biharmonic curves and their properties.

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.

Study of $f$-biharmonic hypersurfaces in conformally flat spaces.

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…

Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.

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.

The paper explores p-biharmonic hypersurfaces in Einstein and conformally flat spaces.

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

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…