Related papers: The codisc radius capacity

Let $\mathcal{M}_{g,\epsilon}$ be the $\epsilon$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,\epsilon}$ is…

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…

We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of…

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the H\"{o}lder sense. This is done by establishing a general trace…

We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate…

Let $M$ be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and $X \subset M$ a compact Lagrangian submanifold. We construct a deformation to the normal cone, showing that a neighbourhood of $X$ can…

We use the generalized Minkowski billiard characterization of the EHZ-capacity of Lagrangian products in order to reprove that the $4$-dimensional Viterbo conjecture holds for the Lagrangian products (any triangle/parallelogram in…

The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics…

We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…

We establish computational results concerning the Lagrangian capacity, originally defined by Cieliebak-Mohnke. More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. Working…

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ \phi_\lambda $ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of…

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for…

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…

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of…

We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our…

The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional…

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer…

For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi self-adjoint tensor field $A$ on it, we use the divergence type operator ${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend some major comparison…

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the…