Praise be to Allah.
In the case of a man, a little adornment is sufficient, such as combing his hair and putting oil in his hair. He is not like a woman who needs to beautify herself.
We do not know what is meant by grooming the eyelashes, because the eyelashes look fine by themselves and do not need any grooming. Allah has created them as an adornment and protection for the eye.
But women use things to thicken the eyelashes, or to make them curl upwards, by using some solutions and cosmetics. What appears to us to be the case, based on what people do, is that this type of adornment is only for women, so it is not permissible for men to imitate them in that, because of the hadith of Ibn ‘Abbaas (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) cursed men who imitate women and women who imitate man. Narrated by al-Bukhaari (5435).
And Allah knows best.