f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970.In this paper,the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-...f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970.In this paper,the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions.The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map.This generalizes the well-known characterization for harmonic morphisms.Some properties and many examples as well as some non-existence of f-harmonic morphisms are given.The author also studies the f-harmonicity of conformal immersions.展开更多
For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classica...For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau （i.e., when ： is constant）.展开更多
In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Ri...In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×（μ,λ） N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.展开更多
We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic m...We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic maps. We showed that some F-harmonic map such as the trigonometric functions instead of the scalar field in the lagrangian, allow, in the absence of term of potential, reproduce the inflation. However, there are other F-harmonic maps such as exponential maps which can’t produce the inflation;the pressure and the density of this exponential harmonic field being both of the same sign. On the other hand, these exponential harmonic fields redraw well the phenomenon of the quintessence when the variation of these fields remains weak. The problem of coincidence, however remains.展开更多
In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Rie...In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.展开更多
In this paper, the author discusses the stable F-harmonic maps, and obtains the Liouville-type theorem for F-harmonic maps into δ-pinched manifolds, which improves the ones in [3] due to M Ara.
We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in&quo...We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in"swapping"comparison theorems in differential equations.These dualities generate comparison theorems on differential equations of mixed typesⅠandⅡ(see Theorems 2.3 and 2.4)and lead to comparison theorems in Riemannian geometry(see Theorems 2.5 and 2.8)with analytic,geometric,PDE's and physical applications.In particular,we prove Hessian comparison theorems(see Theorems 3.1-3.5)and Laplacian comparison theorems(see Theorems 2.6,2.7 and 3.1-3.5)under varied radial Ricci curvature,radial curvature,Ricci curvature and sectional curvature assumptions,generalizing and extending the work of Han-Li-Ren-Wei(2014)and Wei(2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship(see Theorem 5.4).These provide tools in extending the notion,integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms,introducing Condition W for bundle-valued differential forms,and proving the duality theorem and the unity theorem,generalizing the work of Andreotti and Vesentini(1965)and Wei(2020).We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature,generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds,the embedding theorem for weighted Sobolev spaces of functions on manifolds,geometric differential-integral inequalities,generalized sharp Hardy type inequalities on Riemannian manifolds,monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles,such as F-Yang-Mills fields(when F is the identity map,they are Yang-Mills fields),generalized Yang-Mills-Born-Infeld fields on manifolds,Liouville type theorems for Fharmonic maps(when F(t)=1/p(2 t)展开更多
Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extensio...Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps. In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensionM manifolds and some special maps from or into a warped product manifold.展开更多
Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects ...Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects so as to broaden the notions of f-tension field and bi-tension field. We introduce a very large generalization of harmonic maps called f-bi-harmonic maps as the critical points of f-bi-energy functional, and then derive the Euler-Lagrange equation of f-bi-energy functional given by the vanishing of f-bi-tension field.Subsequently, we study some properties of f-bi-harmonic maps between the same dimensional manifolds and give a non-trivial example. Furthermore, we also study the basic properties of f-bi-harmonic maps on a warped product manifold so that we could find some interesting and complicated examples.展开更多
基金supported by the Guangxi Natural Science Foundation(No.2011GXNSFA018127)
文摘f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970.In this paper,the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions.The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map.This generalizes the well-known characterization for harmonic morphisms.Some properties and many examples as well as some non-existence of f-harmonic morphisms are given.The author also studies the f-harmonicity of conformal immersions.
文摘For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau （i.e., when ： is constant）.
基金supported by NSFC (Grant Nos.11971358,11571259,11771339)Hubei Provincial Natural Science Foundation of China (No.2021CFB400)+1 种基金Fundamental Research Funds for the Central Universities (No.2042019kf0198)the Youth Talent Training Program of Wuhan University。
文摘In this article,we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.
基金Partially supported by Guangxi Natural Science Foundation (2011GXNSFA018127)
文摘In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×（μ,λ） N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.
文摘We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic maps. We showed that some F-harmonic map such as the trigonometric functions instead of the scalar field in the lagrangian, allow, in the absence of term of potential, reproduce the inflation. However, there are other F-harmonic maps such as exponential maps which can’t produce the inflation;the pressure and the density of this exponential harmonic field being both of the same sign. On the other hand, these exponential harmonic fields redraw well the phenomenon of the quintessence when the variation of these fields remains weak. The problem of coincidence, however remains.
基金Supported by National Natural Science Foundation of China (Grant No. 10971170)
文摘In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.
文摘In this paper, the author discusses the stable F-harmonic maps, and obtains the Liouville-type theorem for F-harmonic maps into δ-pinched manifolds, which improves the ones in [3] due to M Ara.
基金supported by National Science Foundation of USA(Grant No.DMS1447008)。
文摘We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in"swapping"comparison theorems in differential equations.These dualities generate comparison theorems on differential equations of mixed typesⅠandⅡ(see Theorems 2.3 and 2.4)and lead to comparison theorems in Riemannian geometry(see Theorems 2.5 and 2.8)with analytic,geometric,PDE's and physical applications.In particular,we prove Hessian comparison theorems(see Theorems 3.1-3.5)and Laplacian comparison theorems(see Theorems 2.6,2.7 and 3.1-3.5)under varied radial Ricci curvature,radial curvature,Ricci curvature and sectional curvature assumptions,generalizing and extending the work of Han-Li-Ren-Wei(2014)and Wei(2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship(see Theorem 5.4).These provide tools in extending the notion,integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms,introducing Condition W for bundle-valued differential forms,and proving the duality theorem and the unity theorem,generalizing the work of Andreotti and Vesentini(1965)and Wei(2020).We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature,generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds,the embedding theorem for weighted Sobolev spaces of functions on manifolds,geometric differential-integral inequalities,generalized sharp Hardy type inequalities on Riemannian manifolds,monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles,such as F-Yang-Mills fields(when F is the identity map,they are Yang-Mills fields),generalized Yang-Mills-Born-Infeld fields on manifolds,Liouville type theorems for Fharmonic maps(when F(t)=1/p(2 t)
基金Supported by Science Research Project of Higher Schools of Guangxi(2015ZD019)Key Project of Guangxi University for Nationalities(2012MD033)
文摘Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps. In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensionM manifolds and some special maps from or into a warped product manifold.
基金supported by the Science and Technology Research Project of Guangxi Universities(Grant No.2015ZD038)the Key Scientific Research Project of Guangxi University for Nationalities(Grant No.2012MDZD033)
文摘Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects so as to broaden the notions of f-tension field and bi-tension field. We introduce a very large generalization of harmonic maps called f-bi-harmonic maps as the critical points of f-bi-energy functional, and then derive the Euler-Lagrange equation of f-bi-energy functional given by the vanishing of f-bi-tension field.Subsequently, we study some properties of f-bi-harmonic maps between the same dimensional manifolds and give a non-trivial example. Furthermore, we also study the basic properties of f-bi-harmonic maps on a warped product manifold so that we could find some interesting and complicated examples.
