arXiv research

Paper shows biharmonic Hermitian vector bundles are harmonic on Kaehler Einstein manifolds.

New biharmonic submanifolds found in complex projective spaces.

The paper characterizes biharmonic submersions from product manifolds.

In this paper, we study biharmonic Riemannian submersions. We first derive bitension field of a general Riemannian submersion, we then use it to obtain biharmonic equations for Riemannian submersions with $1$-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. These are used…

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf…

In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$ into the doubly warped product $_{f}B\times_{b}F$ can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections $_{f}B\times_{b}…

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the i…

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of…

The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case th…

In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…

Study ruled real hypersurfaces in nonflat complex space forms with constant norm shape operators.

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…

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 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.

The paper classifies biharmonic immersions and submersions in specific spheres.

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

Reviews recent work on biharmonic conformal maps and their properties.

Totally biharmonic hypersurfaces in space forms and 3D BCV spaces classified.

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.

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…

Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.

Extends $p$-biharmonic and bi-$p$-harmonic map definitions.

Proves Chen's conjecture on biharmonic submanifolds in Euclidean space and space forms.

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 …

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.

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and…

No proper biharmonic CMC compact hypersurface in a specific warped product space.

The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.

Study on biharmonic map heat flow with monotonicity formula.

Extends biharmonic concepts to new hypersurfaces and curves.

Study biharmonic functions on vector bundles with spherical symmetry.

A submanifold $M$ of a Euclidean $m$-space is said to be biharmonic if $Δ\overrightarrow H=0$ holds identically, where $\overrightarrow H$ is the mean curvature vector field and $Δ$ is the Laplacian on $M$. In 1991, the author conjectured that every biharmonic submanifold of a Euclidean space is minimal. The study of b…

The study classifies equivariant biharmonic maps and proves stability results for certain maps.

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.

In this note, we give a brief survey on some recent developments of biharmonic submanifolds. After reviewing some recent progress on Chen's biharmonic conjecture, the Generalized Chen's conjecture on biharmonic submanifolds of non-positively curved manifolds, and some classifications of biharmonic submanifolds of spher…

The paper studies stability and index of biharmonic hypersurfaces in Riemannian manifolds.

The study examines biharmonic hypersurfaces in Sasakian space forms.