Related papers: Weak curvature conditions and functional inequalit…

In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalised) Robertson-Walker space-time is important. In particular, it is a requirement for the development…

We give a sufficient condition for a lightlike isotropic submanifold $M$, of dimension $n$, which is not totally geodesic in a semi-Riemannian manifold of constant curvature $c$ and of dimension $n+p (n < p)$, to admit a reduction of…

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs (and certain higher dimensional inverse systems of metric measure spaces) which imply that the measured Gromov-Hausdorff limit (equivalently, the…

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal…

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or…

We show that the Harnack inequality for a class of degenerate parabolic quasilinear PDE $$\p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu),$$ associated to a system of Lipschitz continuous vector fields $X=(X_1,...,X_m)$ in in $\Om\times (0,T)$…

We establish an optimal transportation inequality for the Poisson measure on the configuration space. Furthermore, under the Dobrushin uniqueness condition, we obtain a sharp transportation inequality for the Gibbs measure on…

We extend the range of $N$ to negative values in the $(K,N)$-convexity (in the sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the…

This work is devoted to the geometric analysis of metric-measure spaces satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb inequality. Completing the early investigations by Cordero-Erausquin, McCann and Schmuckenschlager,…

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also…

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,\mu)$ is a locally complete and separable metric measure space, then continuous functions…

We prove a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The remarkable class of examples of spacetimes…

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds $MCP(0,N)$ and having Euclidean volume growth at infinity. We avoid the classical use of the…

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak…

We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded…

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci…

The measure contraction property, $\mathsf{MCP}$ for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker…

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…