A sharp bound for the area of minimal surfaces in the unit ball

TL;DR

This paper establishes a lower bound for the area of minimal surfaces in the unit ball that meet the boundary orthogonally, confirming a conjecture by R. Schoen and advancing geometric analysis.

Contribution

It provides a sharp lower bound for the area of minimal surfaces in the unit ball, solving a previously open problem posed by R. Schoen.

Findings

The area of such minimal surfaces is at least the volume of the unit k-ball.

The bound is sharp and attained by certain minimal surfaces.

This result advances understanding of boundary behavior of minimal surfaces.

Abstract

Let \Sigma be a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of \Sigma is bounded from below by the volume of the unit ball in R^k. This answers a question posed by R. Schoen.