Let (N, J) be a simply connected 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on N compatible with J to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to i...

We consider second-order linear elliptic operators of nondivergence type which is intrinsically defined on Riemannian manifolds. Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, we give another proof to Harnack inequality of Yau for positive harmonic functions on Riemannian ...

it is easily seen that a motion and a homothetic transformation are both affine collineations and that an affine collineation preserves the curvature tensor. In [l ] one of the present authors proved that in a space of nonzero constant curvature a mapping preserving curvature is a motion. For an Einstein space with nonzero curvature scalar, a mapping preserving Ricci curvature is a motion; for ...

We introduce a new, simple metric method of sampling metric measure spaces, based on a well-known “snowflakeing operator” and we show that, as a consequence of a classical result of Assouad, the sampling of doubling metric spaces is bilipschitz equivalent to that of subsets of some R . Moreover, we compare this new method with two other approaches, in particular to one that represents a direct ...

‎The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces‎. ‎Quasi-Einstein metrics serve also as solution to the Ricci flow equation‎. ‎Here‎, ‎the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined‎. ‎In compact case‎, ‎it is proved that the quasi-Einstein met...

We show that on a compact Riemannian manifold with boundary there exists u ∈ C(M) such that, u|∂M ≡ 0 and u solves the σk-Ricci problem. In the case k = n the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the σk-Ricci problem. By adopting results of [14], we show an interesting relationship between the co...

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace–Beltrami operator on M . We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of...