An orthorhombic deformation family of Schwarz' H surfaces

Abstract

The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS of genus three discovered by W. Meeks, while the H surfaces are among the few known examples outside this family. We construct a 2-parameter family of embedded TPMS of genus three that contains the H family and meets the Meeks family. In particular, we prove that H surfaces can be deformed continuously within the space of TPMS of genus three into Meeks surfaces.

Stacking disorder in periodic minimal surfaces

“…• There exists a real number π/2 > θ * > π/3 such that whenever τ = exp(iθ) with θ < θ * , the configuration is balanced with q 0 = c(1 + τ ) for a unique c < 1/2. This leads to the orthorhombic deformations of Schwarz' H surfaces in [CW18b]. Existence and uniqueness of θ * was essentially proved [Web02,WHW09], and independently in [LW10].…”

“…Existence and uniqueness of θ * was essentially proved [Web02,WHW09], and independently in [LW10]. Its value was computed in [CW18b] explicitly as…”

“…Example 3. Examples with q 0 = −q 1 were studied in [CW18b]. In particular • τ = exp(iπ/3), q 0 = (1 + τ )/3, gives hexagonal Schwarz' H family.…”

“…where m is the unique solution of 2E(m) = K(m), and K(m), K (m) and E(m) are elliptic integrals of the first kind, associated first kind, and second kind, respectively. • For general τ , non-trivial q 0 's that give balanced configurations are studied in [Web02, LW10, CKLW18] and numerically in [CW18b].…”

Existence of the tetragonal and rhombohedral deformation families of the gyroid

“…Moreover, tG and rGL surfaces are not contained in the 5-parameter family of TPMSg3s constructed by Meeks [Mee90]. Today, the only other explicitly known TPMSg3s outside Meeks' family are the recently discovered 2-parameter families oH (containing Schwarz' H) [CW18b] and o∆ [CW18a].…”