Writing your name on the snow is probably a romantic and self-appreciative way, but you don’t want your name to last forever.
The snow is white and flawless, symbolizing purity. The purpose of writing your name on the snow is to express your purity, warmth and romance. Because snow will not stay on the earth for a long time, it will melt and disappear when the weather gets warmer, and the names left in the snow will also disappear naturally. This is exactly the opposite of someone carving "someone came here to visit" in the scenic area. Leave your name in the world forever.