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<p>In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean 3-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another interesting property is that the symmetry group of the genus <i>k</i> surfaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma _{k,t}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the dihedral group with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(4(k+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> elements. Moreover, in particular, for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(|t|=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> we find the family of the Costa–Hoffman–Meeks embedded minimal surfaces, which have two catenoidal ends and a middle flat end. Among the non-embedded examples obtained, there are noncongruent minimal surfaces, with the same symmetry group and conformal structure, as we have in Ramos Batista (Tohoku Math. J. (2) 56(2):237–254, 2004).</p>

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A family of higher genus complete minimal surfaces that includes the Costa–Hoffman–Meeks one

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  • 发表日期：

    2025年5月16日

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  • Irene I. Onnis,
  • Bárbara C. Valério,
  • José Antonio M. Vilhena

摘要

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean 3-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another interesting property is that the symmetry group of the genus k surfaces \(\Sigma _{k,t}\) Σ k , t is the dihedral group with \(4(k+1)\) 4 ( k + 1 ) elements. Moreover, in particular, for \(|t|=1\) | t | = 1 we find the family of the Costa–Hoffman–Meeks embedded minimal surfaces, which have two catenoidal ends and a middle flat end. Among the non-embedded examples obtained, there are noncongruent minimal surfaces, with the same symmetry group and conformal structure, as we have in Ramos Batista (Tohoku Math. J. (2) 56(2):237–254, 2004).