In this work, we aim to prove algebra properties for generalized Sobolev spaces W ∩L on a Riemannian manifold, whereW s,p is of Bessel-typeW s,p := (1+L)(L) with an operator L generating a heat semigroup satisfying off-diagonal decays. We don’t require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another ...

We consider an optimization problem related to mass transportation: given two probabilities f and f− on an open subset Ω ⊂ R , we let vary the cost of the transport among all distances associated with conformally flat Riemannian metrics on Ω which satisfy an integral constraint (precisely, an upper bound on the L-norm of the Riemannian coefficient). Then, we search for an optimal distance which...

theory of Lie groups—but many geometric questions lead to non-standard problems which go beyond, or lie on the frontiers of, standard theory. Such questions arise both in reassuringly down-to-earth topics like the existence of constant mean curvature surfaces in R, and in more esoteric ones such as the use of harmonic maps into 'buildings', which have been exploited by Gromov and Schoen to obta...

The notion the spread of a matrix was first introduced fifty years ago in algebra. In this article, we define the spread of the shape operator by applying the same idea to submanifolds of Riemannian manifolds. We prove that the spread of shape operator is a conformal invariant for any submanifold in a Riemannian manifold. Then, we prove that, for a compact submanifold of a Riemannian manifold, ...

Abstract We obtain new lower bounds for the first non-zero eigenvalue of scalar sub-Laplacian 3-Sasaki metrics, improving Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms automorphism algebra. Our results can be thought as an analogue Lichnerowicz–Matsushima estimate Kähler–Einstein metrics. In dimension 7, if alg...

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We t...

We logically model uncertainty by expanding language without changing logical reasoning rules. We expand the language of set theory by adding new predicate symbols, uncertain membership relations ∈+ and ∈−. We define the set theory ZF± as an extension of ZF with new symbols in classical logic. In this system we can represent uncertainty which is naturally represented in the model of 3-valued lo...

into the rank-3 bundles of self-dual and anti-self-dual 2-forms, respectively defined as the ±l-eigenspaces of the Hodge star operator * : /\ -> / \ ; this just reflects the fact that the adjoint representation of 50(4) on the skew (4 x 4)-matrices is the sum of two 3-dimensional representations, as indicated by the Lie algebra isomorphism so(4) = so(3)©so(3). The decomposition (1) is conformal...