TITLE

# Biharmonic map heat flow into manifolds of nonpositive curvature

AUTHOR(S)
Lamm, Tobias
PUB. DATE
April 2005
SOURCE
Calculus of Variations & Partial Differential Equations;Apr2005, Vol. 22 Issue 4, p421
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
LetMm andNbe two compact Riemannian manifolds without boundary. We consider theL2 gradient flow for the energy. IfandNhas nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.
ACCESSION #
16176841

## Related Articles

• Null 2-type submanifolds of the Euclidean space E5 with non-parallel mean curvature vector. Dursun, Ugur // Journal of Geometry;Apr2007, Vol. 86 Issue 1/2, p73

Let M be a 3-dimensional submanifold of the Euclidean space E5 such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional...

• A variational characterization for sn/2. Brendle, Simon; Viaclovksy, Jeff A. // Calculus of Variations & Partial Differential Equations;Aug2004, Vol. 20 Issue 4, p399

We present here a conformal variational characterization in dimension n = 2k of the equation sk(Ag) = constant, where A is the Schouten tensor. Using the fully nonlinear parabolic flow introduced in, we apply this characterization to the global minimization of the functional.

• Conformally flat 3-t-a manifolds. Gouli-Andreou, F.; Karatsobanis, J.; Xenos, Ph. // Differential Geometry--Dynamical Systems;2008, p107

In this paper, generalizing the results of [5] and [9], we investigate conformally flat 3-Ï„-a manifolds. We find a new class of contact metric manifolds whose non-compact examples have already been constructed by D. E. Blair ([3]) and we find out compact examples. We also give a new example...

• The partial positivity of the curvature in Riemannian symmetric spaces. Liu, Xusheng // Chinese Annals of Mathematics;Jun2008, Vol. 29 Issue 3, p317

In this paper, the partial positivity (resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces is determined. From the classifications of abstract root systems and maximal subsystems, the author gives the calculations for symmetric spaces both in classical types and in...

• On Proper Helices in Pseudo-Riemannian Submanifolds. Hwa Hon Song // Journal of Geometry;2009, Vol. 91 Issue 1/2, p150

Let M be a pseudo-Riemannian submanifold in a pseudo-Riemannian manifold $$\bar{M}$$ isometrically immersed by f, d the order of helix s in M and $$\bar{d}$$ the order of helix f ï¿½ s in $$\bar{M}$$. In specific cases we show the relations among d, $$\bar{d}$$ and M.

• Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case. Yuxin Ge; Xingwang Xu // Calculus of Variations & Partial Differential Equations;Apr2008, Vol. 31 Issue 4, p549

The main objective of this short note is to give a sufficient condition for a non constant function k to be Q curvature candidate for a conformal metric on a closed Riemannian manifold with the null Q-curvature. In contrast to the prescribed scalar curvature on the two-dimensional flat tori, the...

• Moment Maps, Scalar Curvature and Quantization of Kï¿½hler Manifolds. Arezzo, Claudio; Loi, Andrea // Communications in Mathematical Physics;Apr2004, Vol. 246 Issue 3, p543

Building on Donaldsonï¿½s work on constant scalar curvature metrics, we study the space of regular Kï¿½hler metrics E?, i.e. those for which deformation quantization has been defined by Cahen, Gutt and Rawnsley. After giving, in Sects. 2 and 3 a review of Donaldsonï¿½s moment map...

• Differential Geometry of Submanifolds of Warped Product Manifolds I ï¿½ f S m-1( k). Bang-Yen Chen; Shihshu Walter Wei // Journal of Geometry;2009, Vol. 91 Issue 1/2, p21

We provide a general study of submanifolds in R m( k, f) := I ï¿½ f S, which is the warped product of an open interval I and a Riemannian manifold S of constant sectional curvature k. Fundamental properties of submanifolds in Rm( k, f) are obtained. Several classification theorems on...

• 4-DIMENSIONAL ANTI-Kï¿½HLER MANIFOLDS. Kim, H.; Kim, J. // Acta Mathematica Hungarica;2004, Vol. 104 Issue 3, p265

We show that every 4-dimensional anti-Kï¿½hler manifold is Einstein and locally symmetric. In particular any 4-dimensional anti-Kï¿½hler manifold with zero scalar curvature is flat.

Share