Minimal surfaces,Hopf differentials and the Ricci condition |
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Authors: | Theodoros Vlachos |
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Institution: | (1) Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece |
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Abstract: | We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials,
which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf
differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the
study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to
Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces
in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture
is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture
for exceptional minimal surfaces in odd dimensional spheres or in S
4m
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 53A10 Secondary 53C42 |
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