Translating solution of mean curvature flow (MCF) is a self-similar solution that moves by translation under MCF. In the first part, I will survey results regarding translating surface which is graphical and translates in the graphical axis direction (or vertically). In the non-graphical setting, we can WLOG assume the translation direction to be vertical. However for graphical setting, vertical translating direction is equivalent to mean convexity condition whereas for non-vertical translating direction, we might lose mean convexity. In the second part, I will talk about my recent work on the rigidity results of graphical translator moving in non-vertical direction. This is a joint work with Pyo and Ma (preprint arXiv:2210.03707).

 

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