ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT

Paolo Caldiroli; Alessandro Iacopetti; Monica Musso

ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT

Paolo Caldiroli, Alessandro Iacopetti, Monica Musso

Department of Mathematical Sciences

University of Turin

Research output: Contribution to journal › Article › peer-review

1 Citation (SciVal)

Abstract

In Euclidean 3-space endowed with a Cartesian reference system, we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size a and n lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when n is large. Considering a class of mappings H: (Formula presented) (Formula presented) such that H(X) 1 as |X| (Formula presented) with some decay of inverse-power type, we show that for n large and | a| small, in a suitable neighborhood of any Delaunay torus with n lobes and neck-size a there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals H at every point.

Original language	English
Pages (from-to)	193-252
Number of pages	60
Journal	Advances in Differential Equations
Volume	27
Issue number	3-4
Early online date	7 Feb 2022
Publication status	Published - 31 Mar 2022

Bibliographical note

Funding Information:
The first and the second author are members of the Gruppo Nazionale per ??Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The third author has been supported by EPSRC research GrantEP/T008 458/1. This paper has been written when the second author was at D?partement de Math?matique, Universit? Libre de Bruxelles and was also supported by FNRS-F.R.S.

Funding Information:
The third author has been supported by EPSRC research GrantEP/T008 458/1. This paper has been written when the second author was at Départe-ment de Mathématique, UniversitéLibre de Bruxelles and was also supported by FNRS-F.R.S.

ASJC Scopus subject areas

Analysis
Applied Mathematics

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https://projecteuclid.org/journals/advances-in-differential-equations/volume-27/issue-3_2f_4/On-the-non-existence-of-compact-surfaces-of-genus-one/ade027-0304-193.short

Cite this

Caldiroli, P., Iacopetti, A., & Musso, M. (2022). ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT. Advances in Differential Equations, 27(3-4), 193-252. https://projecteuclid.org/journals/advances-in-differential-equations/volume-27/issue-3_2f_4/On-the-non-existence-of-compact-surfaces-of-genus-one/ade027-0304-193.short

Caldiroli, P, Iacopetti, A & Musso, M 2022, 'ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT', Advances in Differential Equations, vol. 27, no. 3-4, pp. 193-252. <https://projecteuclid.org/journals/advances-in-differential-equations/volume-27/issue-3_2f_4/On-the-non-existence-of-compact-surfaces-of-genus-one/ade027-0304-193.short>

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abstract = "In Euclidean 3-space endowed with a Cartesian reference system, we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size a and n lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when n is large. Considering a class of mappings H: (Formula presented) (Formula presented) such that H(X) 1 as |X| (Formula presented) with some decay of inverse-power type, we show that for n large and | a| small, in a suitable neighborhood of any Delaunay torus with n lobes and neck-size a there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals H at every point.",

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N2 - In Euclidean 3-space endowed with a Cartesian reference system, we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size a and n lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when n is large. Considering a class of mappings H: (Formula presented) (Formula presented) such that H(X) 1 as |X| (Formula presented) with some decay of inverse-power type, we show that for n large and | a| small, in a suitable neighborhood of any Delaunay torus with n lobes and neck-size a there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals H at every point.

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ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT

Abstract

Bibliographical note

ASJC Scopus subject areas

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Concentration phenomena in nonlinear analysis

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ON THE NON-EXISTENCE OF COMPACT SURFACES OF GENUS ONE WITH PRESCRIBED, ALMOST CONSTANT MEAN CURVATURE, CLOSE TO THE SINGULAR LIMIT

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Concentration phenomena in nonlinear analysis

Cite this