NONEMBEDDABILITY OF THE KLEIN BOTTLE IN RP 3 AND LAWSON’S CONJECTURE

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In 1985 Montiel & Ros showed that the only minimal torus in S3 , for which the first eigenvalue of the Laplacian is 2, is the Clifford torus. Here, we will show first the nonexistence of an embedded Klein bottle in RP3 . Indeed we will prove that the only non orientable closed surfaces that can be embedded in RP3 are those with odd Euler characteristic. Later on, we will give another proof of Montiel & Ros’ result, assuming that the minimal torus has {x, –x} simmetry.