content_image/370299.jpg content_image/370309.jpg content_image/370310.jpg content_image/370311.jpg content_image/370312.jpg
<img src='content_image/370299.jpg'>

results in Table II. The charges are all normalized by λ , we set n = 1, and all the elliptic integrals have an argument y . 

The analysis for toroidal membranes with a nontrivial potential goes in a similar way. Let us again illustrate it with Case 1. From the periodic boundary condition, we have 

Then the energy is expressed as 

<img src='content_image/370309.jpg'>

<img src='content_image/370310.jpg'>

The results for other cases and different charges are reported in Table III. One notes that Case 1 and Case 3 give identical results for toroidal membranes. 

For both cylindrical and toroidal membranes, it is obvious that one can take the large energy limit in y → 1, where K ( y ) , K (1 /y ) develop a logarithmic divergence. For Case 1 and 3, the normalized charges scale as 

And for Case 2, 

<img src='content_image/370311.jpg'>

<img src='content_image/370312.jpg'>

One can of course do better and expand Δ in terms of J$_{3}$ or Q$_{3}$ , but we relegate the results as well as the details to the appendix. 

