Illinois Journal of Mathematics

The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Minh Hoang Nguyen

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Abstract

In this paper, we study the Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$ with possible infinite boundary data, where $\mathrm{Sol}_{3}$ is the non-Abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins–Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in $\mathrm{Sol}_{3}$.

Article information

Source
Illinois J. Math., Volume 58, Number 4 (2014), 891-937.

Dates
Received: 28 January 2014
Revised: 9 June 2015
First available in Project Euclid: 6 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1446819293

Digital Object Identifier
doi:10.1215/ijm/1446819293

Mathematical Reviews number (MathSciNet)
MR3421591

Zentralblatt MATH identifier
1328.53077

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53A35: Non-Euclidean differential geometry 53B25: Local submanifolds [See also 53C40]

Citation

Nguyen, Minh Hoang. The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data. Illinois J. Math. 58 (2014), no. 4, 891--937. doi:10.1215/ijm/1446819293. https://projecteuclid.org/euclid.ijm/1446819293


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