The card says that: "A monster appears. If you do not defeat it, you are Cursed."

If the investigator runs away, is he still Cursed?" I think so, but I need confirmation.

Sorry, it's a Location card --- not a Mythos card.

Yes, because you haven't defeated it. You essentially have to destroy the monster in order to prevent the curse.

The card says that: "A monster appears. If you do not defeat it, you are Cursed."

If the investigator runs away, is he still Cursed?"

Well, have you defeated that monster before running away?

I think the OP means "Evade" - and yes, the Investigator is Cursed, due to failing to defeat it. As in most cases, each word matters a great deal when determining how to interpret it correctly.

Cheers,

Joe