Note on the growth of Area functions

A Note on the Spectral Area of Toeplitz Operators

In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol.

关于Erdös最小面积问题的一个注记

A note on the Erdös minimal area problem

本文回答了Erdös、Herzog和Piranian关于多项式等值线最小面积的问题，其中多项式的所有零点被限制在紧集K上，且K的对数容量严格大于1。

Minimal surfaces with the area growth of two planes; the case of infinite symmetry

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.

Sharp distortion growth for bilipschitz extension of planar maps

This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.