Breather Surface

A breather surface is a surface obtained from a breather solution of the sine-Gordon equation. At points where its coordinate map satisfies the regular parameterization condition, meaning that the two coordinate tangent vectors are linearly independent, it has constant Gaussian curvature .

The image above, created by Bryant (2026), shows one full period in of the breather surface for , with and . The two long, curved-sided spikes are the tapering ends of the surface. As increases, and tend to 0, causing the colored sheets to crowd toward the x-axis. Along the middle, periodic variation in produces a spring-like sequence of lobes that narrow to necks and flare outward. The apparent stack of tubes consists of overlapping folds of the same parametrized surface, and the cyclic colors make those folds easier to follow.

The construction uses the Gauss-Codazzi equations for a surface of Gaussian curvature , which reduce to the sine-Gordon equation in suitable asymptotic coordinates. The fundamental theorem of surface theory then shows that a solution locally determines a surface in up to a rigid motion. In particular, the breather solutions give a one-parameter family of breather surfaces (Terng and Uhlenbeck 2000; Popov 2014, pp. 168-174).

For , set . A surface corresponding to the associated breather solution has the parametric equations

			(1)
			(2)
			(3)

where is the auxiliary common denominator

(4)

Direct calculation gives the Gaussian curvature

(5)

at every such point (Virtual Math Museum). The singular curves are the images of curves in the -plane along which the two coordinate tangent vectors become linearly dependent. Their occurrence is consistent with Hilbert's theorem on regular surfaces, which rules out a complete regular surface of constant negative Gaussian curvature in (Terng and Uhlenbeck 2000).

The parametric equations contain the two frequencies 1 and , so the surface is periodic in when is a rational number. For example, gives and period . As , the family approaches the Kuen surface up to a rigid motion (Virtual Math Museum).