填充面积猜想、最优收缩不等式和阿贝尔覆盖中的纤维类

Flat currents modulo p in metric spaces and filling radius inequalities

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to provide a proof of Gromov's filling radius inequality (and therefore also the systolic inequality) which applies to nonorientable manifolds, as well. With this goal in mind, we use the Ekeland principle to provide quasi-minimizers of the mass mod(p) in the homology class, and use the…

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.

Discrete surfaces with length and area and minimal fillings of the circle

We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian surfaces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrical…

最优Loewner型不等式中等式的边界情形

Boundary case of equality in optimal Loewner-type inequalities

本文证明了闭黎曼流形（X，g）的某些最优收缩不等式，这些不等式依赖于一对参数n和b.这里n是X的维数，而b是它的第一个贝蒂数。不等式的证明涉及到构造从X到其Jacobi环面T^b的Abel-Jacobi映射，这些映射关于适当的范数是面积减少的（在b维面积上）。这些范数是g的稳定范数、共形不变范数以及其他L^p-范数。等式的情况用T^b的甲板变换格的临界性来表征，而阿贝尔-雅可比映射是调和黎曼浸没。在Abel-Jacobi映射的典型纤维到最大自由阿贝尔覆盖的提升的同调类不为零的假设下，由Federer和Fleming的等周不等式得出的不等式实际上是非空的.