arXiv subjects

Ye-Lin Ou

Publications and source records attributed to Ye-Lin Ou.

At least 19 recordsLinked to original sources

arXiv2026

On biharmonic conformal hypersurfaces

In this paper, we first derive biharmonic equation for conformal hypersurfaces in a generic Riemannian manifold generalizing that for biharmonic hypersurfaces in \cite{Ou1} and that for biharmonic conformal surfaces in \cite{Ou3, Ou2, Ou4}. We then show that if a totally umbilical hypersurface in a space form admits a biharmonic conformal immersion into the ambient space, then the conformal factor has to be an isoparametric function. We also prove that no part of a non-minimal totally umbilical hypersurface in a space form of nonpositive curvature admits a biharmonic conformally immersion into that space form whilst, for the positive curvature space form, we show that the totally umbilical hypersurface $S^4(\frac{\sqrt{3}}{2})\hookrightarrow S^5$ does admit a biharmonic conformal immersion into $S^5$.

math.DG↗

arXiv2024

A short survey on biharmonic Riemannian submersions

The study of biharmonic submanifolds, initiated by B. Y. Chen and G. Y. Jiang independently, has received a great attention in the past 30 years with many important progress. This note attempts to give a short survey on the study of biharmonic Riemannian submersions which are a dual concept of biharmonic submanifolds (i.e., biharmonic isometric immersions).

math.DG↗

arXiv2024

Biharmonic functions and bi-eigenfunctions on some model spaces

In this paper, we first give a convenient formula for bi-Laplacian on a sphere and the complete description of its eigenvalues, buckling eigenvalues, and their corresponding eigenfunctions. We then show that the radial (or rotationally symmetric) solutions for biharmonic equation on the model space $( \mathbb{R}^+\times S^{m-1}, dr^2 + \sigma^2(r)\, g^{S^{m-1}})$ can be given by an integral formula. We also prove that the model space always admits proper biharmonic functions as the products of any eigenfunctions of the factor sphere with certain radial functions. Many explicit examples of proper biharmonic functions on space forms are given. Finally, we give a complete classification of proper biharmonic functions with positive Laplacian on the punctured Euclidean space.

math.DG↗

arXiv2023

Biharmonic Riemannian submersions from a 3-dimensional BCV space

BCV spaces are a family of 3-dimensional Riemannian manifolds which include six of Thurston's eight geometries. In this paper, we give a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional BCV space by proving that such biharmonic maps exist only in the cases of $H^2\times\mathbb{R}\to \mathbb{R}^2$ or $\widetilde{SL}(2,\mathbb{R})\to \mathbb{R}^2$. In each of these two cases, we are able to construct a family of infinitely many proper biharmonic Riemannian submersions. Our results on one hand, extend a previous result of the authors which gave a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional space form, and on the other hand, can be viewed as the dual study of biharmonic surfaces (i.e., biharmonic isometric immersions) in a BCV space studied in some recent literature.

math.DG↗

arXiv2023

Harmonic Riemannian submersions from 3-dimensional geometries

In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.

math.DG↗

arXiv2023

Biharmonic Riemannian submersions from $M^2\times R$

In this paper, we study biharmonic Riemannian submersions $\pi:M^2\times\r\to (N^2,h)$ from a product manifold onto a surface and obtain some local characterizations of such biharmonic maps. Our results show that when the target surface is flat, a proper biharmonic Riemannian submersion $\pi:M^2\times\r\to (N^2,h)$ is locally a projection of a special twisted product, and when the target surface is non-flat, $\pi$ is locally a special map between two warped product spaces with a warping function that solves a single ODE. As a by-product, we also prove that there is a unique proper biharmonic Riemannian submersion $H^2\times \r\to \r^2$ given by the projection of a warped product.

math.DG↗

arXiv2023

Biharmonic isometric immersions into and biharmonic Riemannian submersions from Berger 3-spheres

In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.

math.DG↗

arXiv2023

Harmonic and biharmonic Riemannain submersions from Sol space

In this paper, we give a complete classification of harmonic and biharmonic Riemannian submersions $\pi:(R^3,g_{Sol})\to (N^2,h)$ from Sol space into a surface by proving that there is neither harmonic nor biharmonic Riemannian submersion $\pi:(R^3,g_{Sol})\to (N^2,h)$ from Sol space no matter what the base space $(N^2,h)$ is. We also prove that a Riemannian submersion $\pi:(R,g_{Sol})\to (N^2,h)$ from Sol space exists only when the base space is a hyperbolic space form.

math.DG↗

arXiv2022

Biharmonic homogeneous polynomial maps between spheres

In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit Euclidean sphere $\mathbb{S}^3$. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is $2$, and we obtain the classification of all proper biharmonic quadratic forms from $\mathbb{S}^1$ to $\mathbb{S}^n$, $n \geq 2$, from $\mathbb{S}^m$ to $\mathbb{S}^2$, $m \geq 2$, and from $\mathbb{S}^m$ to $\mathbb{S}^3$, $m \geq 2$.

math.DG↗

arXiv2022

Bi-eigenmaps and biharmonic submanifolds in a sphere

In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps ($Δ^2 ϕ=λϕ$) or buckling eigenmaps ($Δ^2 ϕ=-μΔϕ$). We then classify biharmonic bi-eigenmaps and buckling eigenmaps into spheres with constant energy density. The results can be viewed as generalizations of Takahashi's characterization of minimal submanifolds in a sphere by eigenmaps.

math.DG↗

arXiv2021

Some classifications of conformal biharmonic and k-polyharmonic maps

We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.

math.DG↗

arXiv2020

Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.

math.DG↗

arXiv2019

A note on equivariant biharmonic maps and stable biharmonic maps

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give an improved second variation formula for biharmonic maps into a space form and use it to prove that there exists no stable proper biharmonic maps with constant square norm of tension field from a compact Riemannian manifold without boundary into a space form of positive sectional curvature.

math.DG↗

arXiv2019

Some recent work on biharmonic conformal maps

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, though biharmonic morphisms (maps that preserve solutions of bi-Laplace equations), generalized harmonic morphisms (maps that pull back germs of harmonic functions to germs of biharmonic functions), and biharmonic conformal and Riemannian submersions will also be touched.

math.DG↗

arXiv2019

Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$

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 hypersurfaces in $S^m\times \mathbb{R}$ and $H^m\times \mathbb{R}$ in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi-parallel for $m\ge 3$, and some classifications of biharmonic surfaces in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$ which are constant angle or belong to certain classes of rotation surfaces.

math.DG↗

arXiv2018

Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces

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 and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen's conjecture on biharmonic hypersurfaces by using the biahrmonic graph equation (Theorem 4.1) which paves a way for analytic study of the conjecture.

math.DG↗

arXiv2018

Some remarks on bi-f-harmonic maps and f-biharmonic maps

In this paper, we prove that the class of bi-f-harmonic maps and that of f-biharmonic maps from a conformal manifold of dimension not equal to 2 are the same (Theorem 1.1). We also give several results on nonexistence of proper bi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifolds into nonpositively curved Riemannian manifolds. These include: any bi-f-harmonic map from a compact manifold into a non-positively curved manifold is f-harmonic (Theorem 1.6), and any f-biharmonic (respectively, bi-f-harmonic) map with bounded f and bounded f-bienrgy (respectively, bi-f-energy) from a complete Riemannian manifold into a manifold of strictly negative curvature has rank < 2 everywhere (Theorems 2.2 and 2.3).

math.DG↗

arXiv2018

Biharmonic Riemannian submersions

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 to construct examples of proper biharmonic Riemannian submersions with $1$-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions.

math.DG↗