The phrase "batting eyes at" is typically used to denote someone who is trying to flirt or show interest in another person. However, there are several antonyms for this phrase that can be used to convey the opposite. For example, "avoiding eye contact" suggests shyness or discomfort in engaging with someone in a flirtatious manner, while "ignoring" implies a lack of interest altogether. "Disregarding" suggests a more intentional dismissiveness or even disdain for the person being flirted with. Other potential antonyms for batting eyes at might include "rejecting", "critiquing", or even "mocking". Ultimately, the choice of antonym will depend on the specific context and tone of the situation.