TY - JOUR
T1 - Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group
AU - Young, Robert
N1 - Funding Information:
This material is based upon work supported by the National Science Foundation under Grant Nos. 2005609 and 1926686 and research done while the author was a visiting member at the Institute of Advanced Study. The author would like to thank Sebastiano Nicolussi Golo, Manuel Ritoré, Richard Schwartz, and the anonymous referee for their time and advice during the preparation of this paper and to thank the Institute of Advanced Study for its hospitality.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/8
Y1 - 2022/8
N2 - In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group H to be area-minimizing in the slab { - 1 < x< 1 }. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.
AB - In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group H to be area-minimizing in the slab { - 1 < x< 1 }. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.
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U2 - 10.1007/s00526-022-02264-x
DO - 10.1007/s00526-022-02264-x
M3 - Article
AN - SCOPUS:85131071092
SN - 0944-2669
VL - 61
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 4
M1 - 142
ER -