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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 and Continuous Dynamical Systems, 2015, 35 (10) : 4987-5001. doi: 10.3934/dcds.2015.35.4987
References:
[1]

M. M. Fall and C. Mercuri, Minimal disc-type surfaces embedded in a perturbed cylinder, Differential Integral Equations, 22 (2009), 1115-1124.

[2]

H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114. doi: 10.1007/BF01258269.

[3]

R. Huff and J. McCuan, Scherk-type capillary graphs, J. Mathematical Fluid Mechanics, 8 (2006), 99-119. doi: 10.1007/s00021-004-0140-8.

[4]

R. Lopez and J. Pyo, Capillary surfaces of constant mean curvature in a right solid cylinder, Math. Nachrichten, 287 (2014), 1312-1319. doi: 10.1002/mana.201200301.

[5]

S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map, Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, 1481, Springer, Berlin, 1991, 147-174. doi: 10.1007/BFb0083639.

[6]

F. Morabito, A Costa-Hoffman-Meeks type surface in $\mathbbH^2 \times \mathbbR$, Trans. Am. Math. Soc., 363 (2011), 1-36. doi: 10.1090/S0002-9947-2010-04952-9.

[7]

F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbbR^3$, Boundary Value Problems, (2014), p130. Published online only at: http://www.boundaryvalueproblems.com/content/2014/1/130.

[8]

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

[9]

M. Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk, Annales Institut Fourier, 46 (1996), 1385-1442. doi: 10.5802/aif.1554.

[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, Proceedings of the Clay Mathematical Institute, 2, A co-publication of the AMS and Clay Mathematics Institute, 2005, 19-63.

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), 1115-1124.

[2]

H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114. doi: 10.1007/BF01258269.

[3]

R. Huff and J. McCuan, Scherk-type capillary graphs, J. Mathematical Fluid Mechanics, 8 (2006), 99-119. doi: 10.1007/s00021-004-0140-8.

[4]

R. Lopez and J. Pyo, Capillary surfaces of constant mean curvature in a right solid cylinder, Math. Nachrichten, 287 (2014), 1312-1319. doi: 10.1002/mana.201200301.

[5]

S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map, Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, 1481, Springer, Berlin, 1991, 147-174. doi: 10.1007/BFb0083639.

[6]

F. Morabito, A Costa-Hoffman-Meeks type surface in $\mathbbH^2 \times \mathbbR$, Trans. Am. Math. Soc., 363 (2011), 1-36. doi: 10.1090/S0002-9947-2010-04952-9.

[7]

F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbbR^3$, Boundary Value Problems, (2014), p130. Published online only at: http://www.boundaryvalueproblems.com/content/2014/1/130.

[8]

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

[9]

M. Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk, Annales Institut Fourier, 46 (1996), 1385-1442. doi: 10.5802/aif.1554.

[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, Proceedings of the Clay Mathematical Institute, 2, A co-publication of the AMS and Clay Mathematics Institute, 2005, 19-63.

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