| The content of this paper is divided into two chapters. In the first chapter,we discuess the F-harmonic maps ,which is a generalization of harmonic maps, In the second chapter, we study the metric deformation on compact Riemannian manifolds with boundary.let F : [0, +∞) â†' [0, +∞)be a C2 function,and F' > 0 on (0, +∞). for a smooth map (?) : M â†' N between Riemannian manifolds (M,g) and(N,h) ,Are[1]introduced the notion of F— energy of (?):we call (?) an F-harmonic map if it is a critical point of the F— energy functionalwhen F(t) = t, (2t)p/2/p,et, it is harmonic maps P— harmonic maps and exponentially harmonic maps, respectively .by compute the first variation of F— energy functional , Are get the following resultProposition 1.1.1 [Ara1]: (?) : M â†' N is a .F-harmonic map if and only ifwhereIn[Ara1],Ara studied the second variation of F— energy functional ,later Li .J.T write it as follows (see [Li] )Lemma 1.1.1 [Li]: let (?) : M â†' N be a F— harmonic map, then the second variation formula iswhere V ∈ Γ((?)-1TN),(?) is the connection of Γ((?)-1TN).An F- harmonic map <^> is called F stable or stable if I{V, V) > 0 for any compactly supported vector field V along <ï¿¡If for any V € Fi^TN), else we call ) is the F— energy functional tensor field of We can derive the following important formulas:l.For any vector field X with compact supportf (divSF())(X)+ f (VX, SfW) = 0 (2)Jm Jm2.If dD is a hypersurface in M f F(l-^-)(X,n)= f FJ($Q-)(4>.X,.n)+[ (VX,SF())+[ (divSF())(X) (3)JdD * JdD I JD JDBy use above formulas we can proof :Theorem 1.2.1:let M be a completed simply-connected m-dimensional Riemannian manifold, with nonpositive section curvature and the section curvature on M vary not largely (see the proof for detail ) ,assume be a .F-harmonic map from M to any Riemannian manifold satisfy :xF'(x)<(CFwhere CF = inf{C > O\F'(x)/tc is nonincreasing}if the F- energy of must be constant oTheorem 1.2.2: Let : M —> N be a F-harmonic map, F satisfy :xF'(x) < (Cf + l)-F(z) ' then for any x € Bi (x0) ^D 0 < cr < /) < |,we have the monotonicity formula as follows :T2(CF+l)-mf F(]ffï¿¡!!) * 1 < eCApp2(CF+l)-m fK 2 'where C is a constant only dependent on M,and A is a constant only dependent on the up and below bounded of section curvature in Bi(xq)<,In [Ara2] Ara have derived a Bochner formula ofF-harmonic map,use it we can show:Theorem 1.3.1:Let M be a completely, noncompact Riemmanian manifold with nonnegative Ricci curvature,and N be a iliemmamanmanifold, N is a F—harmonic map with finite F-energy ,assume F satisfy :xF'(x) < (Cf 4- l)F(x) ^P F'(x) + 2xF"(x) > Cf + 1, -i- |V?i<7!?| < C\d must be constant oSet C$ — {x 6 M\d(f>x = 0} M* := M — C^ ,it is wellknow the vertical space at x is define as :VX = ker{dcf>x} € TXM;and the horizontal space at x is Hx — V^-We say horizontally conformal if there exists a smooth function A : M* —* R+ such thatX2g(X, Y) = h(dcf>(X),d(f>(X)) for all X.Y € Vj- and x e M*oIf 0 ,if F : [0 — oo] —> [0 — oo] be a strictly increasing C2 function , then any horizontally conformal F—harmonic map is constant oTheorem 1.4.2: Let M be a completely, noncompact Riemmanian manifold , N be a Riemmanian manifold which assumes exist a strictly subharmonic function /, A/ > 0,assume JM(F'(^-)\dcj)\)2 < oo and E(f) < oo ,or JM F'(^|L)|dc6| < oo and / is bounded , then any horizontally conformal F-harmonic map 4) manifolds, if the initial metric possess positive curvature operator and strong pinching conditions, then we can get the similiar results, consult the reference papers [Hul]In 1996, Shen [Shen] applied Hamilton's Ricci flow to study the metric deformation on Riemannian manifolds with boundary. Shen prove a short time existence theorem for manifolds with umbilical boundary. He also derived the Simons' identity for the boundary under the Ricci flow. And as a corollary, Shen show that any three-manifolds with totally geodesic boundary which admits positive Ricci curvature can be deformed to a space form with totally geodesic boundary.Let (M,g) is an n-dimensional manifold. We adopt the convebtion that Latin indices range from 1 to n, while Greek indices range from 1 to n — 1. suppose that the boundary dM â– =ï¿¡ 0. Let g = {gij} and h = {hap} be the metric of M and the second fundamental form of dM in M. We denote Ricci curvature and scalar curvature as Re = {Rij} and R. It is well known that the Riemannian curvature tensor Rm = {Rijki} of M can be decomposed into three orthogonal components which have the same symmetries as Rm:Rm = W + V + U, (l.l.l)Here W.. = {Wijki} is the Weyl conformal curvature tensor, whereas V = {Vijki} and U = {Uijki} denote the traceless Ricci part and the scalar curvature part respectively. We need the following definition.Definition We say that dM is umbilical in M if the identityhap = \gap, (1.1.2)holds on dM, where A is a constant.In the case A = 0, we say that dM is totally geodesic.Now we are going to state the main result in the paper of [Shen].Theorem [Shen] For any given Riemannian manifolds (M, go), there is a short time solution to the following equations:ï¿¡t9ij = -2IUj, x € M,gij{x,0) = go{x), x G M, (1-1-3)x € dM.Corollary [Shen] Let (M,g) be a compact three-dimensional Riemannian manifold with totally geodesic boundary and with positive Ricci curvature Then (M, g) can be deformed to (M, 4) Riemannian manifold with boundary and get the following result:Definition C(n) — pinched metric we mean R > Oandlijmp < C{n)R2 (1.1.4)where i?and Rm denote the scalar curvature and free curvature tensor respectively. ( see context for detail ) 0Theorem 2.1.1 Let n > 4 , M be a smooth compact n-dimensional Riemannian manifold with totally geodesic boundary carrying a metric with C(n) — pinched curvature , Then (M.g) can be deformed to (M,^) via the Ricci flow such that (M,goo) has constant positive curvature and totally geodesic boundaryD where7,r = 6/25,n = 5 r = 48/125,n =(1.1.5)... |
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