Recently, the Liouville theorem for stable harmonic maps is proved from Euclidean space into sphere. The technique employed in this paper allows the researchers to consider stable harmonic maps from complete manifolds.
By joining two harmonic homogeneous polynomial maps, Smith proved that every element of homotopy group Π<sub>m</sub>（S<sup>m</sup>） has a harmonic representative for m≤7.Now we have proved ...By joining two harmonic homogeneous polynomial maps, Smith proved that every element of homotopy group Π<sub>m</sub>（S<sup>m</sup>） has a harmonic representative for m≤7.Now we have proved the following main result. Every element of odd degree of homotopy group Π<sub>2m+1</sub> (S<sup>2m+1</sup>) has a harmonic representative.Let us consider a function on C<sup>m+1</sup>, due to Nomizu.展开更多
Let M be an m-dimensional complete Riemannian manifold. Denote all the L^2harmonic forms by (?)~*(M). The forms are closed and co-closed by a theorem of Andreotti and Vesentini,
Ⅰ. INTRODUCTION If the field equation of the physical system is derivable from an action functional I, one can proceed to define a so-called stress-energy tensor S, such that at a critical point of I the stress-energ...Ⅰ. INTRODUCTION If the field equation of the physical system is derivable from an action functional I, one can proceed to define a so-called stress-energy tensor S, such that at a critical point of I the stress-energy tensor S is conservative. Recently, P. Baird and J. Eells have successfully introduced such a tensor S for any smooth map between Riemannian manifolds. They showed that harmonic maps satisfy the展开更多
Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map ...Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map from M into N is smooth in the interior of M. If m=d(n)+1, such a map has at most diserete singular set and in general the Hausdorff dimension of the singular set is at most m-d(n)-1.展开更多
文摘Recently, the Liouville theorem for stable harmonic maps is proved from Euclidean space into sphere. The technique employed in this paper allows the researchers to consider stable harmonic maps from complete manifolds.
基金Project supported partially by the National Natural Science Foundation of China and by the Science Foundation of Educational Committee of China.
文摘By joining two harmonic homogeneous polynomial maps, Smith proved that every element of homotopy group Π<sub>m</sub>（S<sup>m</sup>） has a harmonic representative for m≤7.Now we have proved the following main result. Every element of odd degree of homotopy group Π<sub>2m+1</sub> (S<sup>2m+1</sup>) has a harmonic representative.Let us consider a function on C<sup>m+1</sup>, due to Nomizu.
文摘Let M be an m-dimensional complete Riemannian manifold. Denote all the L^2harmonic forms by (?)~*(M). The forms are closed and co-closed by a theorem of Andreotti and Vesentini,
文摘Ⅰ. INTRODUCTION If the field equation of the physical system is derivable from an action functional I, one can proceed to define a so-called stress-energy tensor S, such that at a critical point of I the stress-energy tensor S is conservative. Recently, P. Baird and J. Eells have successfully introduced such a tensor S for any smooth map between Riemannian manifolds. They showed that harmonic maps satisfy the
基金Research at MSRI supported in part by NSF Grant DMS-850550, in part by NNSFC and SFECC.
文摘Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map from M into N is smooth in the interior of M. If m=d(n)+1, such a map has at most diserete singular set and in general the Hausdorff dimension of the singular set is at most m-d(n)-1.
