scholarly journals Perturbations of the Harmonic Map Equation

2003 ◽  
Vol 05 (04) ◽  
pp. 629-669 ◽  
Author(s):  
T. Kappeler ◽  
S. Kuksin ◽  
V. Schroeder
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Homotopy Class ◽  
Harmonic Map ◽  
Poincaré Inequality ◽  
Classical Solutions ◽  
Important Ingredient ◽  
Closed Riemannian Manifold ◽  
Nonpositive Sectional Curvature ◽  
Target Manifold

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

scholarly journals The double cover relative to a convex domain and the relative isoperimetric inequality

2006 ◽  
Vol 80 (3) ◽  
pp. 375-382 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Convex Domain ◽  
Double Cover ◽  
Nonpositive Sectional Curvature ◽  
Relative Isoperimetric Inequality ◽  
Simply Connected

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

scholarly journals On the p-pseudoharmonic map heat flow

2020 ◽  
Vol 31 (13) ◽  
pp. 2050104
Author(s):  
Shu-Cheng Chang ◽  
Yuxin Dong ◽  
Yingbo Han
Keyword(s):  
Riemannian Manifold ◽  
Heat Flow ◽  
Global Existence ◽  
Sectional Curvature ◽  
Compact Riemannian Manifold ◽  
Sasakian Manifold ◽  
Asymptotic Convergence ◽  
Target Manifold

In this paper, we consider the heat flow for [Formula: see text]-pseudoharmonic maps from a closed Sasakian manifold [Formula: see text] into a compact Riemannian manifold [Formula: see text]. We prove global existence and asymptotic convergence of the solution for the [Formula: see text]-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold [Formula: see text] is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the [Formula: see text]-pseudoharmonic map heat flow as well when its initial [Formula: see text]-energy is sufficiently small.

BUBBLING LOCATION FOR SEQUENCES OF APPROXIMATE f-HARMONIC MAPS FROM SURFACES

2010 ◽  
Vol 21 (04) ◽  
pp. 475-495 ◽  
Author(s):  
YUXIANG LI ◽  
YOUDE WANG
Keyword(s):  
Riemannian Manifold ◽  
Riemann Surface ◽  
Critical Points ◽  
Homotopy Class ◽  
Blow Up ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Identity ◽  
Minimal Point ◽  
Closed Riemann Surface

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as [Formula: see text] and its L2-gradient is: [Formula: see text] We will study the blow-up properties of some approximate f-harmonic map sequences in this paper. For a sequence uk : M → N with ‖τf(uk)‖L2 < C1 and Ef(uk) < C2, we will show that, if the sequence is not compact, then it must blow-up at some critical points of f or some concentrate points of |τf(uk)|2dVg. For a minimizing α-f-harmonic map sequence in some homotopy class of maps from M into N we show that, if the sequence is not compact, the blow-up points must be the minimal point of f and the energy identity holds true.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

Ricci-Bourguignon flow coupled with harmonic map flow

2019 ◽  
Vol 30 (10) ◽  
pp. 1950049 ◽  
Author(s):  
Shahroud Azami
Keyword(s):  
Riemannian Manifold ◽  
Existence And Uniqueness ◽  
Harmonic Map ◽  
Coupled System ◽  
Geometric Structures ◽  
Closed Riemannian Manifold ◽  
Harmonic Map Flow

In this paper, we study a coupled system of the Ricci–Bourguignon flow on a closed Riemannian manifold [Formula: see text] with the harmonic map flow. At the first, we will investigate the existence and uniqueness for solution of this flow on a closed Riemannian manifold and then we find evolution of some geometric structures of manifold along this flow.

scholarly journals Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
James Kohout ◽  
Melanie Rupflin ◽  
Peter M. Topping
Keyword(s):  
Constant Curvature ◽  
Harmonic Map ◽  
Gradient Flow ◽  
Minimal Immersion ◽  
Curvature Surface ◽  
Previous Theory ◽  
Target Manifold ◽  
Harmonic Map Flow

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

On the Inner Radius of a Nodal Domain

2008 ◽  
Vol 51 (2) ◽  
pp. 249-260 ◽  
Author(s):  
Dan Mangoubi
Keyword(s):  
Riemannian Manifold ◽  
Lower Bounds ◽  
Type Inequality ◽  
Large Eigenvalue ◽  
Upper And Lower Bounds ◽  
Local Behavior ◽  
Nodal Domain ◽  
Closed Riemannian Manifold ◽  
Inner Radius

AbstractLet M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λα(log λ)β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

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