arXiv subjects

Ze-Ping Wang

Publications and source records attributed to Ze-Ping Wang.

14 recordsLinked to original sources

arXiv2024

$f$-Biharmonic hypersurfaces into a conformally flat space

We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore $f$-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct $f$-biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of $f$-biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate $f$-biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical $f$-biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper $f$-biharmonic $m$-dimensional submanifolds with $m\geq3$ and $m\neq4$ into nonpositvely curved manifolds.

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arXiv2024

Biharmonic conformal immersions into a 3-dimensional conformally flat space

Inspired by the work of Ou [12,17], we study biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first give a characterization of biharmonic conformal immersions of totally umbilical surfaces into a generic 3-manifold. As an application, we give a method to produce biharmonic conformal immersions into a conformally flat 3-space. We then use the method to obtain a classification of biharmonic maps in a family of conformal immersions and construct many examples of biharmonic conformal immersions from a 2-sphere into a conformal 3-sphere. Our examples include proper biharmonic conformal immersions of a 2-sphere minus a point into a conformal 3-sphere with nonconstant conformal factor and the biharmonic isometric immersion $S^2(\frac{1}{\sqrt{2}})\to S^3$ which was found in [2]. Finally, we study biharmonic conformal immersions of Hopf cylinders of a Riemannian submersion.

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arXiv2024

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

In this paper, we study biharmonic Riemannian submersions $π:M^2\times\r\to (N^2,h)$ 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 $π:M^2\times\r\to (N^2,h)$ is locally a projection of a special twisted product, and when the target surface is non-flat, $π$ 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.

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arXiv2024

$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds

$f$-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of $f$-biharmonic curves in a space form. We also obtain a complete classification of proper $f$-biharmonic isometric immersions of a developable surface in $\r^3$ by proving that a proper $f$-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into $\r^3$ exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study $f$-biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper $f$-biharmonic Riemannian submersions and $f$-biharmonic surfaces and curves.

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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.

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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.

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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.

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arXiv2023

Harmonic and biharmonic Riemannain submersions from Sol space

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

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arXiv2015

Biharmonic and f-biharmonic maps from a 2-sphere

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 linear ordinary differential equation. As applications, we give a method to produce biharmonic maps and f-biharmonic maps from given biharmonic maps and we construct many examples of biharmonic and f-biharmonic maps from a round sphere $S^2$ and between two round spheres. Our examples include non-conformal proper biharmonic maps $(S^2, f^{-1}g_0)\longrightarrow S^2$ and $(S^2, f^{-1}g_0)\longrightarrow S^n$, or non-conformal f-biharmonic maps $(S^2, g_0)\longrightarrow S^2$ and $(S^2,g_0)\longrightarrow S^n$ from round sphere with two singular points.

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arXiv2014

Biharmonic maps from tori into a 2-sphere

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 exists any biharmonic map in that homotopy class of maps. In this paper, we obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. Our results show that there exists no proper biharmonic maps of degree $\pm 1$ in a large family of maps from a torus into a sphere.

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arXiv2013

Biharmonic maps from a 2-sphere

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of rotationally symmetric maps between 2-spheres. We also find many examples of proper biharmonic maps defined locally on a 2-sphere. Our results seem to suggest that any biharmonic map $S^2\longrightarrow (N^n, h)$ be a weakly conformal immersion.

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arXiv2011

Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete classifications of constant mean curvature proper biharmonic surfaces in 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classifications of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.

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arXiv2010

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic.

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arXiv2007

Some classifications of \infty-Harmonic maps between Riemannian manifolds

$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between complex Euclidean spaces.

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