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Following similar results in arXiv:1301.5934 for flat tori and round spheres,in this paper is presented a proof of the fact that, for "arbitrary" initialconditions $f_0$, the solution $f_t$ at time $t$ of the heat equation on realor complex projective spaces eventually becomes (and remains) a minimal Morsefunction. Furthermore, it is shown that the solution becomes stable.

Heat equation and stable minimal Morse functions on real and complex projective spaces