GRADIENT ESTIMATES FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS

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）.

f-Harmonic maps of doubly warped product manifolds

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.

Kinetical Inflation and Quintessence by F-Harmonic Map

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.

f-Harmonic Maps Within Bounded Distance from Quasi-isometric Maps

In this article,we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.

Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

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)^(p/2),p>1,they become p-harmonic maps or harmonic maps if p=2),and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and F-harmonic maps(see Theorems 4.1,7.3-7.7,8.1,9.1-9.3,10.1,11.2,12.1 and 12.2),generalizing the work of Caffarelli et al.(1984)and Costa(2008),in which M=R^(n)and its radial curvature K(r)=0,the work of Wei and Li(2009),Chen et al.(2011,2014),Dong and Wei(2011),Wei(2020)and Karcher and Wood(1984),etc.The boundary value problem for bundle-valued differential 1-forms is in contrast to the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton(1975)for the case p=2 and Riem^(N)≤0,and Wei(1998)for 1<p<∞.

Stable P-Harmonic Maps Between Finsler Manifolds

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.