Isometric deformations of the $ {\mathcal{K}}^{\frac{1}{4}}$-flow translators in $ {\mathbb{R}}^{3}$ with helicoidal symmetry

Hojoo Lee

doi:10.1016/j.crma.2013.06.006

Differential Geometry

Isometric deformations of the

K^{\frac{1}{4}}

-flow translators in

R^{3}

with helicoidal symmetry

Presented by: the Editorial Board

Hojoo Lee ¹

¹ Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 477-482

Abstract (Original language)
Résumé

The height functions of $K^{\frac{1}{4}}$ -flow translators in the Euclidean space $R^{3}$ solve the classical Monge–Ampère equation $f_{x x} f_{y y} - f_{x y}^{2} = 1$ . We explicitly and geometrically determine the moduli space of all helicoidal $K^{\frac{1}{4}}$ -flow translators, which are generated from planar curves by the action of helicoidal groups.

Les fonctions de hauteur des translateurs du flot $K^{1 / 4}$ de $R^{3}$ résolvent lʼéquation de Monge–Ampère classique $f_{x x} f_{y y} - f_{x y}^{2} = 1$ . Nous déterminons de manière géométrique explicite lʼespace des modules de tous les translateurs à symétrie hélicoïdale du flot $K^{1 / 4}$ , qui sont engendré à partir de courbes planes par lʼaction de groupes hélicoïdaux.

Received: 2012-07-04
Accepted: 2013-06-19
Published online: 2013-06-01

DOI: 10.1016/j.crma.2013.06.006

Author's affiliations:

Hojoo Lee ¹

¹ Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea

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     title = {Isometric deformations of the $ {\mathcal{K}}^{\frac{1}{4}}$-flow translators in $ {\mathbb{R}}^{3}$ with helicoidal symmetry},
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     pages = {477--482},
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Hojoo Lee. Isometric deformations of the $ {\mathcal{K}}^{\frac{1}{4}}$-flow translators in $ {\mathbb{R}}^{3}$ with helicoidal symmetry. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 477-482. doi: 10.1016/j.crma.2013.06.006

References
Cited by

[1] S.J. Altschuler; L.F. Wu Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, Volume 2 (1994) no. 1, pp. 101-111

[2] B. Andrews Contraction of convex hypersurfaces by their affine normal, J. Differential Geom., Volume 43 (1996) no. 2, pp. 207-230

[3] B. Andrews Gauss curvature flow: the fate of the rolling stones, Invent. Math., Volume 138 (1999) no. 1, pp. 151-161

[4] B. Andrews Motion of hypersurfaces by Gauss curvature, Pacific J. Math., Volume 195 (2000) no. 1, pp. 1-34

[5] E. Calabi Hypersurfaces with maximal affinely invariant area, Amer. J. Math., Volume 104 (1982) no. 1, pp. 91-126

[6] B. Chow Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom., Volume 22 (1985) no. 1, pp. 117-138

[7] J. Clutterbuck; O.C. Schnürer; F. Schulze Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations, Volume 29 (2007) no. 3, pp. 281-293

[8] M.P. do Carmo; M. Dajczer Helicoidal surfaces with constant mean curvature, Tohoku Math. J. (2), Volume 34 (1982) no. 3, pp. 425-435

[9] S. Donaldson A generalised Joyce construction for a family of nonlinear partial differential equations, J. Gökova Geom. Topol. GGT, Volume 3 (2009), pp. 1-8

[10] J. Eells The surfaces of Delaunay, Math. Intelligencer, Volume 9 (1987) no. 1, pp. 53-57

[11] W.J. Firey Shapes of worn stones, Mathematika, Volume 21 (1974), pp. 1-11

[12] G. Haak On a theorem by do Carmo and Dajczer, Proc. Amer. Math. Soc., Volume 126 (1998) no. 5, pp. 1547-1548

[13] H.P. Halldorsson Helicoidal surfaces rotating/translating under the mean curvature flow, Geom. Dedicata, Volume 162 (2013), pp. 45-65

[14] R. Harvey; H. Lawson Calibrated geometries, Acta Math., Volume 148 (1982), pp. 47-157

[15] R. Harvey; H. Lawson Split special Lagrangian geometry | arXiv

[16] K. Jörgens Uber Die Losungen der Differentialgleichung $r t - s^{2} = 1$ , Math. Ann., Volume 127 (1954), pp. 130-134

[17] M. Koiso The Delaunay surfaces, Bull. Kyoto Univ. Ed. Ser. B, Volume 97 (2000), pp. 13-33

[18] H.B. Lawson Complete minimal surfaces in $S^{3}$ , Ann. of Math. (2), Volume 92 (1970), pp. 335-374

[19] H. Lee Minimal surface systems, maximal surface systems and special Lagrangian equations, Trans. Amer. Math. Soc., Volume 365 (2013) no. 7, pp. 3775-3797

[20] J. Mealy Volume maximization in semi-Riemannian manifolds, Indiana Univ. Math. J., Volume 40 (1991), pp. 793-814

[21] M. Spivak A Comprehensive Introduction to Differential Geometry, vol. IV, Publish or Perish, Inc., Wilmington, NC, 1979

[22] N. Trudinger; X.-J. Wang The affine Plateau problem, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 253-289

[23] K. Tso Deforming a hypersurface by its Gauss–Kronecker curvature, Comm. Pure Appl. Math., Volume 38 (1985) no. 6, pp. 867-882

[24] J. Urbas Complete noncompact self-similar solutions of Gauss curvature flows. I. Positive powers, Math. Ann., Volume 311 (1998) no. 2, pp. 251-274

Cited by Sources:

^☆ This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology) [NRF-2011-357-C00007].

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