Tight surfaces with boundary

Abstract

In Differential Geometry an immersed smooth surface in 3-space is called tight if it has the minimum total absolute curvature among all immersed surfaces with the same topology. Tight surfaces without boundary were extensively studied, and tight surfaces with at least two boundary components are known to exist. An announcement of White in 1974 (‘Minimal total absolute curvature for orientable surfaces with boundary’, Bull. Amer. Math. Soc. 80, 361–362) stated that there are no smooth tight orientable surfaces of genus at least one with exactly one boundary component. Here we disprove this statement by a family of counterexamples, starting with a smooth tight torus with one hole. The non-orientable case is also studied. We show that there is no smooth tight Möbius band in 3-space and that there are smooth tight non-orientable surfaces of higher genus with exactly one boundary component. Three non-orientable cases of low genus remain open.