Bounded Harmonic Functions and Positive Ricci Curvature.
Canonical metric on the domain of discontinuity of a kleinian group
Certain contact metrics satisfying the Miao-Tam critical condition
We study certain contact metrics satisfying the Miao-Tam critical condition. First, we prove that a complete K-contact metric satisfying the Miao-Tam critical condition is isometric to the unit sphere . Next, we study (κ,μ)-contact metrics satisfying the Miao-Tam critical condition.
Characterization of Low Dimensional RCD*(K, N) Spaces
In this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse...
Classification of -dimensional homogeneous weakly Einstein manifolds
Y. Euh, J. Park and K. Sekigawa were the first authors who defined the concept of a weakly Einstein Riemannian manifold as a modification of that of an Einstein Riemannian manifold. The defining formula is expressed in terms of the Riemannian scalar invariants of degree two. This concept was inspired by that of a super-Einstein manifold introduced earlier by A. Gray and T. J. Willmore in the context of mean-value theorems in Riemannian geometry. The dimension is the most interesting case, where...
Classification of 4-dimensional homogeneous D'Atri spaces
The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition is said to be of type . The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers...
Closed geodesics in simply connected Riemannian spaces of negative curvature.
Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature.
Compact manifolds of nonnegative isotropic curvature and pure curvature tensor.
Complexifications of nonnegatively curved manifolds
Confomal deformations on a noncompact Riemannian manifold.
Conformal metrics with constant -curvature.
Conformal scalar curvature on the hyperbolic space.
Conformally bending three-manifolds with boundary
Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant....
Conformally geodesic mappings satisfying a certain initial condition
In this paper we study conformally geodesic mappings between pseudo-Riemannian manifolds and , i.e. mappings satisfying , where are conformal mappings and is a geodesic mapping. Suppose that the initial condition is satisfied at a point and that at this point the conformal Weyl tensor does not vanish. We prove that then is necessarily conformal.
Conjugate and cut loci of a two-sphere of revolution with application to optimal control
Constant Jacobi osculating rank of
In this paper we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold . As a consequence, an explicit expression of the Jacobi operator independent of the geodesic can be obtained on such a manifold. Moreover, we show the way to calculate the Jacobi vector fields on this manifold by a new formula valid on every g.o. space.
Constant scalar curvature metrics on connected sums.
Construction of compact constant mean curvature hypersurfaces with topology
In this paper, we explain how the end-to-end construction together with the moduli space theory can be used to produce compact constant mean curvature hypersurfaces with nontrivial topology. For the sake of simplicity, the hypersurfaces we construct have a large group of symmetry but the method can certainly be used to provide many more examples with less symmetries.
Construction of Kähler surfaces with constant scalar curvature