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Harmonic Maps with Potential and Hyper-surface in the Ricci Symmetric Riemannian Manifold

Author: ZhengZuo
Tutor: CaiKaiRen
School: Hangzhou Normal University
Course: Basic mathematics
Keywords: p-H- harmonic maps F - harmonic maps Radial curvature Constant Mean Curvature Parallel mean curvature Hessian comparison theorem Parallel Ricci Lorentz space de Sitter space
CLC: O186.12
Type: Master's thesis
Year: 2009
Downloads: 25
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Abstract


This article studies with geopotential harmonic map and some submanifold geometry, the content is divided into five parts and the first part of the study with potential harmonic maps. Mainly use this part of the Hessian comparison theorem, research radial curvature is non-positive Riemannian manifold with geopotential harmonic maps Constant Boundary and stability problems, the p-harmonic maps with geopotential proved several Liouville type theorem. detailed below Theorem 2.1 Let M be an m-dimensional diameter positive to the curvature of the non-complete simply connected Riemannian manifold, m> 3.γ (t): [0, r] → M starting from p ∈ M regular geodesic γ (t), where p points conjugated to meet along γ radial Ricci curvature of M is not greater than 2 times the radial curvature of M. located. f: M → N is a potential H Riemannian manifold N harmonic maps in measured Earth the B of p (R), and R y ∈ N H (y), then f B p (R) on must be mapped to a constant value. Theorem 2.2 Let M be the m-dimensional radial curvature non-positive complete simply connected without conjugate points of a Riemannian manifold, m> 3.γ starting from p ∈ M formal geodesic meet the radial Ricci curvature of M along γ is not greater than 2 times the radial curvature of M Let f: M → N is a finite energy or energy slow divergence harmonic maps then f must be constant mapping. Theorem 2.3 Let M m-dimensional radial curvature of non-positive complete simply connected Riemannian manifold, m> 3.γ (t): [0, r] → M from p ∈ M starting the regular geodesic, wherein γ (t) on the conjugate point of the point p, and meet radially Ricci curvature along γ of M is not greater than 2 times the radial curvature of M Let f: M → N is pH - harmonic maps, measured Earth B p (R), R y ∈ N H (y), f B p (R) must be a constant value mapping. Theorem 2.4 Let M be a m-dimensional radial curvature not being complete simply connected no conjugate points Riemannian manifold the m> 3.γ is a regular geodesic starting from p ∈ M, meet along γ radial Ricci curvature of M is not greater than 2 times the radial curvature of M Let f: M → N is a finite p-energy slow divergence of the p-energy p-harmonic maps, then f must literal mapping. Theorem 2.5 to set M n space type (?) is compact submanifold If you meet the nc> pc (?) (n> p ≥ 2), then M n to any Riemannian manifold with potential H stable p-harmonic maps must be constant mapping (?) , B ei, ej = B ij u e u , H = B ii u e u . Theorem 2.6 set M n space type (?) in the submanifold satisfied nc> pc (? ) (n> p ≥ 2), and (?) H ≥ 0, any compact Riemannian manifold N to M n with the very values ??of geopotential H p-harmonic maps must is unstable, which (?), B ei, ej = B ij u e u H = B ii u e u . curvature of the second part of the study the number of for being a closed Riemannian manifold manifold as the starting F-harmonic maps. As we all know, from a Ricci curvature positive closed Riemann flow shaped into a sectional curvature non-positive complete Riemannian flow shape is not exist nonconstant harmonic mapping of. as the promotion of harmonic maps, we study a class of a wider range of F-harmonic mapping, to out the curvature of a number of positive closed Riemann flow shaped into a sectional curvature non-positive complete Riemannian flow shaped exists very value of the F-harmonic mapping of conditions. follows Theorem 3.1 Let M m dimension has positive scalar curvature of closed Riemann flow shaped N is a complete Riemannian manifold of non-positive sectional curvature Let φ: M → N is very value of F-harmonic mapping F: [0, ∞) → [0, ∞), F ∈ C 2 , and satisfies F '> 0, F''> 0, then where E = (?), R is the scalar curvature of M. equality holds: R is the universal covering manifold of constant and Riemannian manifold M (?) isometric to R × M ', M' Einstein manifold of scalar curvature R is the normal number of the second part of the study Lorentz space N 1 n 1 1 n 1 the spacelike hypersurface (c), if M is Ricci parallel locally M = M n (c) M = R × M s (c 1 ) x M the ns-1 (c 2 ), or M = M s ( c 1 ) × M ns (c 2 ), which the M n (c), M s (the c 1 ), etc. are constant curvature manifolds. compact hypersurface with constant mean curvature in the fourth part of the study parallel Ricci curvature Riemannian manifold. gives a the J.Simons type integral inequality, promotion of local symmetric spaces such hypersurface the results the following conclusions Theorem 5.1 Let N n 1 parallel Ricci curvature δ-pinched Riemannian manifold M n N n 1 in compact hypersurface with constant mean curvature, H M n Constant Mean Curvature, S for its second fundamental The form of the length of the square, is the establishment of the the integral inequality fifth part of the study of de Sitter space with parallel mean curvature of the n-dimensional complete spacelike manifolds with parallel mean curvature in the de Sitter space n dimensional Complete submanifolds a rigidity theorem. the following conclusions Theorem 6.1 Let M n is the de Sitter space S p np (c) with parallel mean curvature n dimensional complete spacelike manifold if the square of the length of the second fundamental form satisfies S <(?), then M n is totally umbilical, M n is located S p np (c) a totally geodesic submanifold S 1 n 1 (c) .

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry > Riemannian geometry
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