Manifold Bounded Distortion Mapping And Harmonic Maps Studies

This PH.D thesis focus on the geometry and analysis aspects of mappingsof bounded distortions between manifolds. The classical Schwarz-Pick Lemmaand Liouville's Theorem have been generalized to mappings between manifoldswhich satisfy certain conditions. Some Schwarz and Liouville type theorems ofmappings of bounded distortions from a manifold whose curvature is boundedfrom below to the target which is negatively curved have been studied thoroughly.Our theorems could be regarded as the extension of Liouville's theorem for thepositively curved targets. When working on it, we used the famous Bochnertechniques. Then we generalized Bochner formulas to Hermitian manifolds andget some analyticity of harmonic maps.Chapter 1 of this thesis offers preliminaries of our work, began with thedefinition of bounded distortion on geometry and analysis. Since we have knownthe relationship between quasiconformality and bounded distortion for diffeomorphisms which means they have been always studied together, we give out abrief introduction of conformal mappings, G-conformal linear transformations,and harmonic equations for weakly K-quasi regular mappings. Then we introduce the development of bounded distortion, especially for generalized Schwarz'lemma and Liouville's theorem recently, and mentioned Liouville's theorem forweakly quasi regular mappings. Then we describe some knowledge of complexgeometry, which will be used in this thesis. Curvatures of Hermitian complexmanifolds, curvatures of Hermitian vector bundles and induced connections andcurvatures are among them.In Chapter 2, we mainly discuss mappings of bounded s-distortion betweenmanifolds, and try to find if there exist Liouville type theorems of these mappings. The mappings we mainly study are defined by: Definition 0.0.1 A smooth mapping f:(M,g)→(N,h) between orientedn-dimensional manifolds is said to have bounded s-distortion (0＜s＜∞) withrespect to the metric g and h if it is a constant or a local diffeomorphism andfor some positive constant K.The theory of above mappings was born in the x-y plane, which is calledbounded distortion. It is not difficult to see the relationships between boundeddistortion and quasi-regularity, quasirconformality and bounded distortion fordiffeomorphisms. It was in the smooth case that planar quasiconformal mappingswere first studied by Gr(?)tzsch (see []) in 1928. The term "quasiconformal" wasstudied by Ahlfors in 1935(see [], [], []) and used as an integral tool in his re-search of Nevanlinna theory which was based on the "length-area" method. Theywere also successfully used by Drasin in his solution to the inverse Nevanlinnaproblem (see []). Meanwhile, Teichm(?)ller found a fundamental connectionsbetween quasiconformal mappings and quadratic differentials when dealing withextremal mappings between Riemann surfaces (see []) around 1939.Liouville proved the first important theorem about conformal mappings indimensions n≥3 in 1850 (see []).Theorem 0.0.2 Given a domainΩin Rⁿ and a diffeomorphismf:Ω→Rⁿ,we say f is conformal if Df(x) =λ(x)O(x), where for each x∈Ω,λ(x) is scalarand O(x) is an orthogonal n×n matrix.This is a very strong rigidity theorem, of course, not optimal. People aretrying to relax the injectivity assumption or the differentiability assumption asmuch as possible. The natural setting for Liouville's theorem is in the Sobolevspaces, for example W_loc^1,p(Ω,Rⁿ)(1≤p≤∞), where some related definitions weregiven out, including mappings that are weak K-quasiregular, K-quasiregular andK-quasiconformal. The basic connection between quasiregular mappings and non-linear PDEsis Beltrami system, whose matrix could be seen as a metric in geometry. Inthis way, people started to generalize mappings of bounded distortion in geometry. The equavalence of geometric and analytic definition was shown by Gehringand Lehto in 1959 (see []). Moreover, the connection between quasiconformalmappings, Teichm(?)ller theory and quadratic differentials has been intensivelyinvestigated by Ahlfors and Bers, Reich and Strebel , Lehto( see [], [], [])and the references therein.By geometric function theory, T. Iwaniec and G. Martin generalized Liouville's theorem (see [])as follows:Theorem 0.0.3 Every weakly 1- quasiregular f of Sobolev class W_loc^?(Ω,R^?)with (?)＞1, is either constant or the restriction toΩof a M(?)bius transformationof R^?The above theorem holds for even dimensional manifolds, for normal dimensional manifolds in smooth cases, the following theorem holds:Theorem 0.0.4 Every solution f∈C³(Ω,Rⁿ),n≥3, to the Cauchy-Riemannsystem D^tf(x)Df(x)= |J(x,f)|^2/nI,J(x,f) does not change sign inΩ, has theformhere, a∈Rⁿ,b∈Rⁿ are arbitrary parameters,α∈R , A is a orthogonal matrix,and (?) is either 0 or 2.Another interesting result about conformal mappings is Schwarz lemma. Inthis field, our works start by the following classical Schwarz-Pick lemma(see []): Theorem 0.0.5 Let D =D={z∈C‖z|＜1}. The Poincare-Bergman metric onD is given byIf we let d_D(x, y) denote the corresponding distance functions, then any holomorphic map f:D→D is distance decreasing with respect to Poincare-Bergmanmetric, i.e.,In 1937, Ahlfors found that the Gaussian curvature remains invariant underconformal mappings in complex plane, and generalized Schwarz lemma to holomorphic mappings between two Riemann surfaces whose curvatures were usedin a very explicit way (see []). Later, Chern initiated the study of holomorphicmappings between higher-dimensional complex manifolds by generalizing Ahlfors'lemma (see []). After that, this lemma was further extended by S. Kobayashi,Griffiths, Wu, Lu and others, and plays a very important role in their theorems.In 1970, P.J.Kiernan investigated K quasiconformal mappings of Riemannsurfaces(see []). A theorem, which is similar to Schwarz's lemma, is proved fora certian class of K quasiconformal mappings. This result is then used to giveelementary proofs of known theorems concerning K quasiconformal mappings including Liouville's theorem. He used a method of Bochner techniques (see []),computed a formula for the LaplacianΔ‖df‖² which involves the curvatures ofM and M', and implied that under some additional assumptions, harmonic Kquasiconformal mappings are distance decreasing.In 1978, A major advance was Yau's Schwarz's lemma. The main point ofhis result is that the domain M is a very general manifold. Methods employedpreviously in proving Schwarz lemma depend largely on a nice requirement ofthe manifold M. Yau eliminated these hypotheses by applying a method thatdeveloped before, that is, the gradient, laplacian of a existed point of a C² function which is bounded from below on a complete Riemannian manifold with Ricci curvature bounded from below are required(see []).Yau's proof is extremely useful in differential geometry and complex analysis(see []). Later generalizations of his result were mainly in two directions:relaxing the curvature hypothesis or the K(?)hler assumption (see [], [], [])orproving similar results for harmonic maps of Riemannian manifolds(see [] ).There are also several generalizations of Yau's Schwarz lemma to the casesof almost Hermitian manifolds. The significant contributions were done by Z.Chen, H. Yang, S.K.Donaldson, D.P.Sullivan([], [],[],[],[]) and recently byV. Tosatti([]). The Schwarz lemmas between Riemannian manifolds were foundby S. I. Goldberg, Z. Har'el, T. Ishihara, N. C. Petridis([],[], [], []), andC. L. Shen([]).Theorem 0.0.6 [Generalized Schwarz Lemma] Let M, N be complete Riemannian manifolds, the Ricci curvature of M is bounded below by-K₁ and thesectional curvature of N is bounded from above by-K₂ where K₁,K₂＞0. Iff:M→N is a harmonic K-quasiregular mapping, thenwhere C is a positive constant depending on K and the dimension of the manifolds.The following is a generalized Liouvile's theorem by generalized Schwarz lemma:Theorem 0.0.7 [Generalized Liouville's Theorem] Let N be an n-dimensionalRiemannian manifold with negative sectional curvature bounded away from zero,and let f:R^m→N be a harmonic K-quasiregular mapping, then f is constant.The common conditions in the above Schwarz and Liouville type theoremsare: the curvature of the domain should be bounded from below; the target should be negatively curved; the mapping should satisfy certain bounded distortion condition. Naturally, we were wondering if there are similar theoremsfor positively curved targets instead of negativity. The idea was inspired by M.Troyanov and S. Vodop'yanov who asked a question in their paper (see []): Whatare the obstruction to the existence of a non-constant mapping with bounded sdistortion f:M→N? As many geometers expect, they are the curvaturesof the manifolds. We answer the authors' question and get some results in (see[]):Compared with theorem 0.0.7, if we replace domain R^m by complex manifoldCⁿ and consequently harmonic by holomorphic, then we could get the followinggeneralized Liouville's theorem if the target manifold satisfies certain curvatureconditions:Theorem 0.0.8 Let (N, h) be a complete K(?)hler manifold of complex dimension n, and f:Cⁿ→N be holomorphic. If f is a mapping with bounded2s-distortion and N satisfies the curvature condition (Q_s), then f is constant.Here the curvature property (Q_s) of a complete K(?)hler manifold (N, h) meansfor any nonzero vector field X = (?), we haveIn fact, the curvature condition has a geometric explanation. It is equivalent tothe Griffiths positivity of vector bundle G =(?). We know for anycompact K(?)hler manifold with c₁(M)＞0, the anti-canonical line bundle K_N^* isa positive line bundle, so there exist some small s∈(0, n) such that the vectorbundle G is Griffiths positive, that is the curvature condition Q_s could be satisfied. Let s₀ be the least upper bound of such s. It is obvious that s₀ dependsonly on the manifold N.Theorem 0.0.9 Let M be a compact K(?)hler manifold with complex dimensionn and c₁(M)＞0. Then there exists a K(?)hler metricωand some s₀∈(0, n) suchthat any holomorphic mapping f:Cⁿ→(M,ω)with bounded 2s-distortion, 0＜s＜s₀, is constant.Since for a compact K(?)hler manifold with complex dimension n and c₁(M)＞0, there exists a K(?)hler metricωand some s∈(0, n) such that (M,ω) satisfiesthe curvature condition (Q_s). Then we could get theorem 0.0.9 easily by theorem0.0.8.Corollary 0.0.10 If f:Cⁿ→Pⁿ is a holomorphic mapping with bounded2s-distortion, 0＜s＜(?), with respect to the canonical metrics, then f isconstant, and (?) is accurate.Since the complex projective space Pⁿ is a compact and connected Kahlermanifold, which means it is complete. And when 0＜s＜(?), Pⁿ satisfiesthe curvature condition (Q_s) with respect to Fubini-Study metricω_FS. So bytheorem 0.0.8, f is constant.To get the upper bound of s, we choose a special casef:Cⁿ→Pⁿ, f(z₁,…,z_n)=[1,z₁,…,z_n]. By fundamental computation, we find that / is bounded n + 1-distortion with respect to the Euclidean metricω_C=(?) on Cⁿ andFubini-Study metircω_FS on Pⁿ.In the proof of theorem 0.0.8, we used the famous Bochner techniques .Inthe third Chapter of this thesis, we generalize them to the case of Hermitian com-plex(possibly non-holomorphic) vector bundles over compact Hermitian manifoldwith any metric connection(more general than the Levi-Civita connection) on thetangent bundle as follows:As applications, we could use it to prove the analyticity of harmonic mapsbetween curved manifolds. The most valuable part of our technique is that wecould deal with the positively curved target. We have(see [])Theorem 0.0.11 If (?) is a metric connection on the holomorphic tangent bundle T^1,0M of a Hermitian manifold (M,ω) and (?) is a metric connection on the Hermitian complex vector bundle E, then we havewhereΛ_ωadjoint operator of L(·) =ωΛ·with respect to the L² metric on(M,ω) and (?) is the (1,1) part of the curvture (?).Bochner formula was first given by Bochner which was defined on Rieman-nian manifolds, and later similar formulas defined on compact K(?)hler manifoldsand compact K(?)hler-Einstein manifolds (see []) were given. To prove this theorem, first we know that a metric connection (?) on the holomorphic tangent bundleT^1,0M of a Hermitian manifold (M,ω) has a decomposition (?), andwe could extend the covariant differential operator to (p, q) forms. Then thereexists a natural decomposition (?), and consequently Bochner formulaon Hermitian manifolds. Moreover, if (?) is a symmetric metric connection, thenwe haveAs an application of the formulas, we get the following analyticity of harmonic maps(see []):Theorem 0.0.12 Let f:(N,ω_N)→(M,ω_M) be a harmonic map between acompact K(?)hler manifold (N,ω_N) and a compact hermitian manifold (M,ω_M).If det T^1,0N(?)f^*(T^1,0M) is semi-Nakano positive and positive at some points,then f is holomorphic.First f is harmonic implies (?). Then by theorem 0.0.11 and the expansion of (?), since T^1,0N(?)f^*(T^1,0M) is semi-Nakano, we have(?)0. At last since T^1,0N(?)f^*(T^1,0M) is positive at some point, then f is holomorphic and Aronszajn's principles, we have (?).The following is the famous rigidity result of Siu. He proved it by analyzingthe eigenvalues of the harmonic functions which was very complicated, and we could highly simplify it by generalized Bochner formulas (see []):Theorem 0.0.13 Let f:(N,ω_N)→(M,ω_M) be a harmonic map betweencompact K(?)hler manifolds. If the curvature tensor of M is strongly semi-negativeand strongly negative at f(P) for some point P∈N with rank_Rdf＞4 at P,then f is holomorphic or anti-holomorphic.