A diameter bound for compact surfaces and the Plateau-Douglas problem

TL;DR

This paper establishes a diameter bound for compact surfaces with boundary in Euclidean space, linking it to Topping's bound for closed surfaces, and applies it to the Plateau-Douglas problem to derive a new nonexistence criterion.

Contribution

It introduces a geometric diameter bound for surfaces with boundary, extending Topping's bound and providing a novel criterion for the Plateau-Douglas problem.

Findings

Derived an explicit diameter estimate for surfaces with boundary

Connected the bound to Topping's conjecture for minimal surfaces

Provided examples illustrating the criterion's distinctiveness

Abstract

In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The obtained estimate is expected to be optimal for minimal surfaces in the sense that optimality follows if the Topping conjecture holds true. Our result directly implies an explicit nonexistence criterion in the classical Plateau-Douglas problem. We exhibit examples of boundary contours to ensure that our criterion is of different type from classical criteria based on the maximum principle and White's criterion based on a density estimate.