What is going on with these lines on my cheeks under my eyes?

In my experience, malar creases and tear trough problems do not come about overnight. Quite possibly your recent illness accentuated a process that may have been developing slowly. If in another few weeks the creases remain, fillers, particularly Belotero Balance (in my Upper East Side Manhattan office), would make a good and simple treatment approach. Belotero has little or no tendency to engender the unwanted bluish Tyndall Effect when injected superficially into the delicate tissues below the eyes. In my Israel satellite facility where a far greater number of regulatory agency approved fillers and volumizers are available I would opt for Teosyal Redensity II for this purpose.

The lines that are visible down from the eyes to one's cheeks are ligaments that are more visible especially when one loses volume. This could be due to weight loss, and aging. With strategic injections using Dermal Fillers like Voluma and Radiesse can help revolumize and bring back the apples of your cheeks making you look and feel rejuvenated. 

you have the typical signs of aging, which occurs at different times for different people/genetics.  I suggest a formal cosmetic consultation.  it may be best for you to start mild laser/peel treatments, some fillers, and better skin care to help prevent the process from rapidily progressing.  Best, Dr. Emer.