Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal this paper is to understand notions convergence the structure limits for such spaces. Even in seemingly rigid case $\epsilon_i\to 0$, we construct examples showing that may converge wildly Gromov-Hausdorff or Intrinsic Flat sense. ...

We answer the following question dating back to J. E. Littlewood (1885–1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is to avoid pathological examples where the man survives forever because ...

We extend a discrepancy bound of Lagarias and Pleasants for local weight distributions on linearly repetitive Delone sets show that similar holds also the more general case without finite complexity if linear repetitivity is replaced by $\varepsilon$-linear repetitivity. As result we establish are some sufficiently small $\varepsilon$ rectifiable, incommensurable multiscale substitution tilings...

In this paper, we obtain a unified characterization of uniformly rectifiable sets {\it any codimension} in terms Carleson estimate on the second derivatives Green function. When restricted to domains with boundaries codimension 1, our result generalizes previous Azzam for Laplacian more general elliptic operators. For greater than is completely new.

Abstract In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on rectifiable open curve. The deal considers influence requirement solutions problem, degree non-smoothness curve at endpoints as well behavior coefficient these points.

The classical solution of the Plateau problem by Radó [10] and Douglas [3] shows that any rectifiable Jordan curve in R is spanned by a minimal surface of disc type. Under what conditions a minimal surface of a given higher genus exists, spanning a given Jordan curve in a Riemannian manifold N, seems to be a much more difficult problem. For compact minimal surfaces without boundary and in case ...