(color online) (a) Heat source and heat sink with the temperature T₁ and T₂ are connected by a catenoidal wire with a rectangular cross-sectional area A ( x ) = hl ( x ) , where h is thickness in the z direction and l ( x ) = cosh 2 ( x / λ ) . (b) The total conductance divided by temperature K/T reduced by the zero-temperature universal value π 2 k B 2 / 3 h . The solid, dashed, dotted, and dashed-dotted curves correspond to d 1 = d 2 = 50 nm ; d₁=60, λ=5021.6 nm; d₁=200, λ=1653 nm; and d₁=2000, λ=860 nm, respectively. Here, the d₂=50, a=2176.9, and h = 50 nm. (c) and (d) correspond to the thermal conductance for the lowest six types of vibrational modes divided by temperature K/T reduced by the zero-temperature universal value π 2 k B 2 / 3 h . The solid, dashed, dotted, and dashed-dotted in (c) and (d) correspond to dilatational mode, torsional mode, and two flexural modes in z and y direction, respectively. The solid and dashed curves in inserted figure in (c) and (d) correspond to two shear modes in z and y directions, respectively. The structure parameters in (c) and (d) are d₁=2000, λ=860 nm; d₁=60, λ=5021.6 nm, respectively. Here, d₂=50, a=2176.9, and h = 50 nm. Reprinted with permission from Ref. [113]. Copyright (2010) by American Physical Society.