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Abstract
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.
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References
F. Almgren, some interior regularity theorems for minimal surfaces and an extension of Bernstein theorem, Ann. of Math. 85 (1966), 277-292.
S.S. Chern, M. Do Carmo, And S. Kobasyashi, Minimal submanifolds of a sphere with second fudamental form of constant length. In Functional Analysis and related Fields. Proc. Conf. M. Stone. Springer: Berlin, 1970, 59-75.
M. Do. Carmo, Riemannian Geometry. Birkhauser: Boston, Second edition, 1993.
S. Montiel & A. Ros, Minimal imersion of surfaces by the first eigenfunctions and conformal área. Invent. Math. 83 (1986), 153-166. H. Samelson, Orientability of Hypersurfaces in Rn. Proc. A.M.S. 22(1969), 301-302.
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