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Higher Genus Maxfaces with Enneper End

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Abstract

We have proven the existence of new higher-genus maxfaces with Enneper end. These maxfaces are not the companions of any existing minimal surfaces, and furthermore, the singularity set is located away from the ends. The nature of the singularities is systematically investigated.

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References

  1. Estudillo, F.J.M., Romero, A.: Generalized maximal surfaces in Lorentz-Minkowski space \(l^3\). Math. Proc. Camb. Philos. Soc. 111(3), 515–524 (1992)

    Article  Google Scholar 

  2. Fujimori, S., Mohamed, S.G., Pember, M.: Maximal surfaces in Minkowski 3-space with non-trivial topology and corresponding CMC 1 surfaces in de Sitter 3-space. Kobe J. Math. 33(1–2), 1–12 (2016)

    MathSciNet  Google Scholar 

  3. Fujimori, S., Rossman, W., Umehara, M., Yamada, K., Yang, S.-D.: New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de sitter 3-space. Results Math. 56(1), 41 (2009)

    Article  MathSciNet  Google Scholar 

  4. Imaizumi, T., Kato, S.: Flux of simple ends of maximal surfaces in \({{R}}^{2,1}\). Hokkaido Math. J. 37(3), 561–610 (2008)

    Article  MathSciNet  Google Scholar 

  5. Kim, W., Yang, S.-D.: A family of maximal surfaces in Lorentz-Minkowski three-space. Proc. Am. Math. Soc. 134(11), 3379–3390 (2006)

    Article  MathSciNet  Google Scholar 

  6. Kumar, P., Mohanty, S.R.R.: Approximating singularities by a cuspidal edge on a maxface. Archiv der Mathematik. https://doi.org/10.1007/s00013-022-01747-9 (2022)

  7. Kumar, P., Mohanty, S.R.R.: Genus zero complete maximal maps and maxfaces with an arbitrary number of ends. Rep. Math. 361, 1683–1690 (2023)

    MathSciNet  Google Scholar 

  8. Umehara, M., Yamada, K.: Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35(1), 13–40 (2006)

    Article  MathSciNet  Google Scholar 

  9. Umehara, Masaaki, Yamada, Kotaro: Applications of a completeness lemma in minimal surface theory to various classes of surfaces. Bull. Lond. Math. Soc. 43(1), 191–199 (2011)

    Article  MathSciNet  Google Scholar 

  10. Weber, M., Wolf, M.: Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs. Geom. Funct. Anal. (1998). https://doi.org/10.1007/s000390050125

    Article  MathSciNet  Google Scholar 

  11. Weber, M., Wolf, M.: Teichmuller theory and handle addition for minimal surfaces. Ann. Math. 156, 713–795 (1998)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shiv Nadar Institute of Eminence, Deemed to be University, Dadri, Uttar Pradesh, 201314, India

    Rivu Bardhan, Indranil Biswas & Pradip Kumar

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  1. Rivu Bardhan
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  2. Indranil Biswas
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  3. Pradip Kumar
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Correspondence to Pradip Kumar.

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Bardhan, R., Biswas, I. & Kumar, P. Higher Genus Maxfaces with Enneper End. J Geom Anal 34, 207 (2024). https://doi.org/10.1007/s12220-024-01661-2

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  • DOI: https://doi.org/10.1007/s12220-024-01661-2

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