Negative exterior curvature foliations on compact Riemannian manifolds

The topological structure of compact Riemannian manifolds that admit hyperbolic foliations is studied. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 673–676, November, 1997. Translated by S. S. Anisov     

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L ² harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L ² harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic manifold can not admit any almost K?hler structure of positive isotropic curvature.     

Finslerian foliations of compact manifolds are Riemannian

We prove that a Finslerian foliation of a compact manifold is Riemannian.     

Complete Riemannian manifolds with pointwise pinched curvature

An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M ⁿ . We prove that if n≥4 and the curvature operator of M ⁿ is pointwise pinched, or if n=3 and the Ricci curvature of M ³ is pointwise pinched, then M ⁿ is compact. Oblatum 4-II-1999 & 10-XI-1999?Published online: 21 February 2000     

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

Some compact conformally flat manifolds with non-negative scalar curvature

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.     

Rigidity of Riemannian manifolds with positive scalar curvature

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the \(L^{\frac{n}{2}}\)-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.     

Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

Complete submanifolds in manifolds of partially non-negative curvature

We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.     

Closed hypersurfaces with prescribed Weingarten curvature in Riemannian manifolds

In this paper we consider the problem of finding closed hypersurfaces with prescribed Weingarten curvature in a large class of Riemannian manifolds. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and standard degree theory for the admissible solutions to the prescribed curvature equation. We mainly show that the primary technique developed in Urbas, Wang and the second author’s work in 2004 is flexible enough to be used in more general ambient geometry.     

Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds

We consider certain semi-linear partial differential inequalities on complete connected Riemannian manifolds and provide a simple condition in terms of volume growth for the uniqueness of a non-negative solution. We also show the sharpness of this condition.