arXiv research

Study on proper-biharmonic flat tori in spheres with CMC conditions.

The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.

The paper classifies biharmonic immersions and submersions in specific spheres.

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 …

CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessar…

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

Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there …

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.

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

Reviews recent work on biharmonic conformal maps and their properties.

The paper classifies biharmonic hypersurfaces with constant scalar curvature.

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

Abstract: Characterizations for submersions to be harmonic or biharmonic shown; examples provided.

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…

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

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…

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

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

The paper studies biharmonic submersions in Riemannian geometry.

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

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

The study proves unique continuation for biharmonic maps.

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.

The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.

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

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 biharmonic real hypersurfaces in complex projective spaces.

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.

The paper studies stability and index of biharmonic hypersurfaces in Riemannian 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 study examines biharmonic hypersurfaces in Sasakian space forms.

Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into $(\mathbb{R}^2, σ^2dwd\bar w)$ is always biharmonic if the conformal factor $σ$ is bi-a…

Integral formulas help prove rigidity of biharmonic submanifolds.

In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…

Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.

The paper finds conditions for biharmonic orbits in symmetric spaces.