Related papers: Volume entropy and rigidity for RCD-spaces

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing…

We show that the existence of a nontrivial Massey product in the cohomology ring H^*(X) imposes global constraints upon the Riemannian geometry of a manifold X. Namely, we exhibit a suitable systolic inequality, associated to such a…

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian…

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq…

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its…

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff…

We study compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces with variable exponents on both bounded and unbounded metric measure spaces. We establish sufficient conditions for compactness, and under additional assumptions, we…

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of…

For the Newtonian \(N\)-body problem at nonnegative energy, we study solution sets selected by the Jacobi--Maupertuis variational principle and by the associated stationary Hamilton--Jacobi equation. We prove a compactness/stability theorem…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex…

We prove in this note that local geometric uniqueness holds true without loss of regularity for Einstein equations coupled with a large class of matter models. We thus extend the Planchon-Rodnianski uniqueness theorem for vacuum spacetimes.…

We prove the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes. More precisely, we show that solutions to the Einstein vacuum equations with positive cosmological constant arising from data on a cylinder that is…

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem…

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper,…

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of…

In this paper, we investigate a conformally flat and complete manifold $(M,g)=(\mathbb{R}^n,e^{2u}|dx|^2)$ with finite total Q-curvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that…

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such…

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--\'Emery (via energy and \Gamma_2$-calculus) in complete generality for infinitesimally Hilbertian metric…

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming…