Surfaces of small diameter with large width

Abstract

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Stephane Sabourau.

arXiv comment15 pages, 5 figures. Improved exposition, corrected an error in the statement of Theorem 1.1, added an example due to Florent Balacheff. To appear in Journal of Topology and Analysis