I am not satisfied with the meanings I've found for grilles on the palm. Porbably because they are predominately  negative, but also because I've found nothing relating it's meanings with lines forming it.

Someone whose palm I read said they had another palm reader tell me that a grille on the mount of venus relates to sexual magnetic energy. Which to me makes sense as many worry lines especially when they are faint relates to helping other deal with their worries as one person could not possibly think of that many things to worry about and them being faint mean that the energy does not effect them in a deep way, while the lines circling the bottom of the line are family ties and bonds, so both of these relate to a level of magnetism.