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October  2015, 35(10): 4987-5001. doi: 10.3934/dcds.2015.35.4987

Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$

1. 

KAIST, Department of Mathematical Sciences, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, South Korea

Received  August 2014 Revised  February 2015 Published  April 2015

The aim of this work is to show the existence of free boundary minimal surfaces of Saddle Tower type which are embedded in a vertical solid cylinder in $\mathbb{R}^3$ and invariant with respect to a vertical translation. The number of boundary curves equals $2l$, $l \ge 2$. These surfaces come in families depending on one parameter and they converge to $2l$ vertical stripes having a common vertical intersection line. Such surfaces are obtained by perturbing the symmetrically modified Saddle Tower minimal surfaces.
Citation: Filippo Morabito. Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4987-5001. doi: 10.3934/dcds.2015.35.4987
References:
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M. M. Fall and C. Mercuri, Minimal disc-type surfaces embedded in a perturbed cylinder,, Differential Integral Equations, 22 (2009), 1115.   Google Scholar

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H. Karcher, Embedded minimal surfaces derived from Scherk's examples,, Manuscripta Math., 62 (1988), 83.  doi: 10.1007/BF01258269.  Google Scholar

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R. Huff and J. McCuan, Scherk-type capillary graphs,, J. Mathematical Fluid Mechanics, 8 (2006), 99.  doi: 10.1007/s00021-004-0140-8.  Google Scholar

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R. Lopez and J. Pyo, Capillary surfaces of constant mean curvature in a right solid cylinder,, Math. Nachrichten, 287 (2014), 1312.  doi: 10.1002/mana.201200301.  Google Scholar

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S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map,, Global Differential Geometry and Global Analysis, (1481), 147.  doi: 10.1007/BFb0083639.  Google Scholar

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F. Morabito, A Costa-Hoffman-Meeks type surface in $\mathbbH^2 \times \mathbbR$,, Trans. Am. Math. Soc., 363 (2011), 1.  doi: 10.1090/S0002-9947-2010-04952-9.  Google Scholar

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F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbbR^3$,, Boundary Value Problems, (2014).   Google Scholar

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F. Pacard, Connected sum constructions in geometry and non-linear analysis,, Lecture notes available from: , ().   Google Scholar

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M. Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk,, Annales Institut Fourier, 46 (1996), 1385.  doi: 10.5802/aif.1554.  Google Scholar

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M. Weber, Classical minimal surfaces in euclidean space by examples: Geometric and computational aspects of the Weierstrass representation,, in Global Theory of Minimal Surfaces, (2005), 19.   Google Scholar

show all references

References:
[1]

M. M. Fall and C. Mercuri, Minimal disc-type surfaces embedded in a perturbed cylinder,, Differential Integral Equations, 22 (2009), 1115.   Google Scholar

[2]

H. Karcher, Embedded minimal surfaces derived from Scherk's examples,, Manuscripta Math., 62 (1988), 83.  doi: 10.1007/BF01258269.  Google Scholar

[3]

R. Huff and J. McCuan, Scherk-type capillary graphs,, J. Mathematical Fluid Mechanics, 8 (2006), 99.  doi: 10.1007/s00021-004-0140-8.  Google Scholar

[4]

R. Lopez and J. Pyo, Capillary surfaces of constant mean curvature in a right solid cylinder,, Math. Nachrichten, 287 (2014), 1312.  doi: 10.1002/mana.201200301.  Google Scholar

[5]

S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map,, Global Differential Geometry and Global Analysis, (1481), 147.  doi: 10.1007/BFb0083639.  Google Scholar

[6]

F. Morabito, A Costa-Hoffman-Meeks type surface in $\mathbbH^2 \times \mathbbR$,, Trans. Am. Math. Soc., 363 (2011), 1.  doi: 10.1090/S0002-9947-2010-04952-9.  Google Scholar

[7]

F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbbR^3$,, Boundary Value Problems, (2014).   Google Scholar

[8]

F. Pacard, Connected sum constructions in geometry and non-linear analysis,, Lecture notes available from: , ().   Google Scholar

[9]

M. Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk,, Annales Institut Fourier, 46 (1996), 1385.  doi: 10.5802/aif.1554.  Google Scholar

[10]

M. Weber, Classical minimal surfaces in euclidean space by examples: Geometric and computational aspects of the Weierstrass representation,, in Global Theory of Minimal Surfaces, (2005), 19.   Google Scholar

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