Related papers: Avoiding \sigma-porous sets in Hilbert spaces

We show that a sectional-hyperbolic attracting set for a H\"older-$C^1$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these…

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K…

For negative-torsion maps on the annulus we show that on every $\mathcal{C}^1$ essential curve there is at least one point of zero torsion. As an outcome, we deduce that the Hausdorff dimension of the set of points of zero torsion is…

Let $\Sigma$ be a hyperbolic surface. We study the set of curves on $\Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $\gamma_0$. For example, in the particular case that $\Sigma$ is a…

Let $\tilde{\Sigma}$ be the universal cover of a closed surface $\Sigma$ of genus at least $2$. We characterize all equivariantly area-minimizing maps from $\tilde{\Sigma}$ to a Hilbert sphere, which are equivariant with respect to an…

We study the bounded negativity conjecture for non-quaternionic Hilbert modular surfaces and give an explicit bound for the special case of Hirzebruch-Zagier curves on Hilbert modular surfaces.

We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere $\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel…

A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane…

Let us denote $\lambda$ the Lebesgue measure on $[0,1]$, put$$ C(\lambda)=\{f\in C([0,1]);\ \forall~A\subset [0,1], A~\text{Borel}:\ \lambda(A)=\lambda(f^{-1}(A))\}.$$ We endow the set $C(\lambda)$ by the uniform metric $\rho$ and…

In this note we construct a measure $\mu$ on a $\sigma$-algebra $\mathcal{M}$ of subsets of the positive real axis, $\mathbb{R}_{>0}$, with the following multiplicative property: \[ \mu \left( \bigcup_j E_j \right) = \prod_j \mu(E_j) \] for…

We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this…

We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a…

We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.

Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity \delta_\Omega of a set \Omega. We also show that \lim_{\epsilon \to 0}…

In this paper we prove that the cone $\PPD$ of positive, positive definite, discrete and strong almost periodic measures has an interesting property: given any positive and positive definite measure $\mu$ smaller than some measure in…

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…

In this short note a new proof of the monotone con- vergence theorem of Lebesgue integral on \sigma-class is given.

In the paper we study closures of classes of log--concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$…