The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

Some Results of Biharmonic Maps

In this paper, we investigate biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain some non-existence results for these maps.     

Applications harmoniques, applications pluriharmoniques et existence de 2-formes parallèles non nulles

In this paper we study harmonic maps from a compact riemannian manifold equiped with a non trivial parallel 2-form, to a K?hler manifold of strongly negative curvature tensor or a riemannian manifold of strictly negative complex sectional curvature. In a first part we set up some rigidity results of Siu type. Then we obtain an upper bound for the rank of such maps in terms of the rank of the 2-form and deduce some vanishing theorems. Received: March 4, 1996     

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2 × V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

The coarse geometric Novikov conjecture for subspaces of non-positively curved manifolds

In this paper, we prove the coarse geometric Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature.     

正拼挤流形的F-调和映射

设M~n(n≥3)是R~(n+1)中紧致凸超曲面,本文证明了:若F″≤0且M的n个主曲率λ_i满足0<λ_i<1/2∑_(j=1)~nλ_j,则M~n和任何紧致黎曼流形之间的稳定F-调和映射必为常值映射.     

Complete submanifolds of manifolds of negative curvature

In this article, we give sufficient conditions for a complete non-compact submanifold in a simply connected manifold of negative sectional curvature to have only one end. We also prove some Liouville theorems for harmonic maps from the same kind of submanifolds to manifolds of non-positive curvature.     

Haken 3-Manifolds in Small Covers*

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.     

Manifolds with pinched 2-positive curvature operator

In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and ??-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by ??·scal > 0. If we relax the restriction of positivity of complex sectional curvature to nonnegativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.     

Local Existence for Inhomogeneous Schrödinger Flow into Kähler Manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schr?dinger flow for maps from a compact Riemannian manifold M with dim(M) ≤ 3 into a compact K?hler manifold (N, J) with nonpositive Riemannian sectional curvature Received November 1, 1999, Revised January 14, 2000, Accepted March 29, 2000     

The rigidity theorem for harmonic maps from Berwald manifolds

In this paper, we can prove that any non‐degenerate strongly harmonic map ? from a compact Berwald manifold with nonnegative general Ricci curvature to a Landsberg manifold with non‐positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson ([2]).     

Harmonic map heat flow with free boundary

In this paper, we study the harmonic map heat flow with free boundary from a Riemannian surface with smooth boundary into a compact Riemannian manifold. As a consequence, we get at least one disk-type minimal surface in a compact Riemannian manifold without minimal 2-sphere.     

Holomorphically projective transformations on complex Riemannian manifold

The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.Dedicated to N.K. Stephanidis on the occasion of his 65 th birthday.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Let be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if is homeomorphic to a K?hler manifold, then its Euler number satisfies the inequality . Received March 14, 1998 / Revised August 7, 2000 / Published online December 8, 2000     

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012)  [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.     

Curvature of the space of positive Lagrangians

A Lagrangian submanifold in an almost Calabi–Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature.