March/April 2022 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
Adv. Differential Equations 27(3/4): 193-252 (March/April 2022). DOI: 10.57262/ade027-0304-193

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 $H\colon\mathbb{R}^{3}\to\mathbb{R}$ such that $H(X)\to 1$ as $|X|\to\infty$ 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.

Citation

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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." Adv. Differential Equations 27 (3/4) 193 - 252, March/April 2022. https://doi.org/10.57262/ade027-0304-193

Information

Published: March/April 2022
First available in Project Euclid: 7 February 2022

Digital Object Identifier: 10.57262/ade027-0304-193

Subjects:
Primary: 53A05 , 53A10 , 53C21 , 53C42

Rights: Copyright © 2022 Khayyam Publishing, Inc.

Vol.27 • No. 3/4 • March/April 2022
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