Related papers: Localization and Tensorization Properties of the C…

Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally…

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the…

The aim of this paper is threefold. We first prove that, on $\mathrm{RCD}(K,N)$ spaces, the boundary measure of any set with finite perimeter is concentrated on the $n$-regular set $\mathcal{R}_n$, where $n\le N$ is the essential dimension…

As a continuation of \cite{NSY:local}, we mainly discuss the global structure of two-dimensional locally compact geodesically complete metric spaces with curvature bounded above. We first obtain the result on the Lipschitz homotopy…

We prove higher summability and regularity of \Gamma(f) for functions f in spaces satisfying the Bakry-\'Emery condition BE(K,\infty). As a byproduct, we obtain various equivalent weak formulations of BE(K,N) and we prove the…

Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the Riemannian Curvature Dimension condition, $RCD^*(K,N),$ and a compact subgroup of isometries $G \leq Iso(X)$ we prove that there exists a $G-$invariant measure,…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of…

The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional…

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 \emph{non-empty} one dimensional regular sets. In particular, we prove…

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexsandrov, we show that a non-collapsed $\mathrm{RCD}(0,n)$ space ($n\geq2$) with Euclidean…

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a…

In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the ${\rm CD}(K, m)$-condition for…

We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, $RCD^*(K,N),$ is in fact a Lie group. We obtain an optimal upper bound on its dimension and classify the spaces where this…

Let $({M},\textsf{d},\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\textsf{CD}(K,n)$ for some $K\geq 0$ and $n\geq 2$, and a lower $n-$density assumption at some point of $M$.…

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$,…

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points…

Let $(M,\mathsf{d},\mathfrak{m},\ll,\leq,\tau)$ be a causally closed, $\mathscr{K}$-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition $\smash{\mathrm{wTCD}_p^e(K,N)}$…

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on…

Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is…

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is…