no. 1
Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow
[Applications de la dualité entre l’équation de Monge–Ampère complexe homogène et le flot de Hele-Shaw.]
Ross, Julius1 ; Nyström, David Witt1
1 DPMMS, Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB (UK)
Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 1-30.

Nous donnons deux applications de la dualité entre l’équation de Monge–Ampère complexe homogène (HCMA) et le flot de Hele-Shaw. D’abord nous prouvons l’existence de données lisses au bord pour lesquelles la solution faible au problème de Dirichlet pour l’équation HCMA sur ℙ 1 ×𝔻 ¯ n’est pas deux fois différentiable en certains points fixés a priori ainsi que des exemples qui ne sont pas différentiables le long d’un ensemble de codimension 1 de ℙ 1 ×𝔻 ¯. Puis nous expliquons comment obtenir explicitement des familles de rayons géodésiques lisses dans l’espace des métriques Kähler sur ℙ 1 et sur le disque unité 𝔻. Ils sont construits à partir d’une famille à la fois exhaustive et croissante de domaines simplement connexes variant de manière lisse.

We give two applications of the duality between the Homogeneous Complex Monge–Ampère Equation (HCMA) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HCMA over ℙ 1 ×𝔻 ¯ is not twice differentiable at a given collection of points, and also examples that are not twice differentiable along a set of codimension one in ℙ 1 ×∂𝔻. Second, we discuss how to obtain explicit families of smooth geodesic rays in the space of Kähler metrics on ℙ 1 and on the unit disc 𝔻 that are constructed from an exhausting family of increasing smoothly varying simply connected domains.

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DOI : 10.5802/aif.3237
Classification : 32W20, 76D27
Keywords: Complex Monge–Ampère equations, Hele-Shaw flows
Mot clés : l’équation de Monge–Ampère complexe, flot de Hele-Shaw

Ross, Julius 1 ; Nyström, David Witt 1

1 DPMMS, Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB (UK)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ross, Julius; Nyström, David Witt. Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow. Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 1-30. doi : 10.5802/aif.3237. https://aif.centre-mersenne.org/articles/10.5802/aif.3237/

[1] Bedford, Eric; Demailly, Jean-Pierre Two counterexamples concerning the pluri-complex Green function in C n , Indiana Univ. Math. J., Volume 37 (1988) no. 4, pp. 865-867 | DOI | MR | Zbl

[2] Bedford, Eric; Taylor, Bert A. The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44 | DOI | MR | Zbl

[3] Błocki, Zbigniew The C 1,1 regularity of the pluricomplex Green function, Mich. Math. J., Volume 47 (2000) no. 2, pp. 211-215 | DOI | MR | Zbl

[4] Błocki, Zbigniew On geodesics in the space of Kähler metrics, Advances in geometric analysis (Advanced Lectures in Mathematics (ALM)), Volume 21, International Press, 2012, pp. 3-19 | MR | Zbl

[5] Chen, Xiuxiong The space of Kähler metrics, J. Differ. Geom., Volume 56 (2000) no. 2, pp. 189-234 http://projecteuclid.org/euclid.jdg/1090347643 | DOI | MR | Zbl

[6] Darvas, Tamás Morse theory and geodesics in the space of Kähler metrics, Proc. Am. Math. Soc., Volume 142 (2014) no. 8, pp. 2775-2782 | DOI | MR | Zbl

[7] Darvas, Tamás; Lempert, László Weak geodesics in the space of Kähler metrics, Math. Res. Lett., Volume 19 (2012) no. 5, pp. 1127-1135 | DOI | MR | Zbl

[8] Donaldson, Simon Kirwan Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar (Advances in the Mathematical Sciences), Volume 196, American Mathematical Society, 1999, pp. 13-33 | DOI | MR | Zbl

[9] Donaldson, Simon Kirwan Holomorphic discs and the complex Monge–Ampère equation, J. Symplectic Geom., Volume 1 (2002) no. 2, pp. 171-196 http://projecteuclid.org/euclid.jsg/1092316649 | DOI | MR | Zbl

[10] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, xiv+517 pages (Reprint of the 1998 edition) | MR | Zbl

[11] Gustafsson, Björn; Vasilʼev, Alexander Conformal and potential analysis in Hele-Shaw cells, Advances in Mathematical Fluid Mechanics, Birkhäuser, 2006, x+231 pages | MR | Zbl

[12] Hedenmalm, Håkan; Olofsson, Anders Hele-Shaw flow on weakly hyperbolic surfaces, Indiana Univ. Math. J., Volume 54 (2005) no. 4, pp. 1161-1180 | DOI | MR | Zbl

[13] Hedenmalm, Håkan; Shimorin, Sergei Hele-Shaw flow on hyperbolic surfaces, J. Math. Pures Appl., Volume 81 (2002) no. 3, pp. 187-222 | DOI | MR | Zbl

[14] Krantz, Steven G. Function theory of several complex variables, American Mathematical Society, 2001, xvi+564 pages (Reprint of the 1992 edition) | DOI | MR | Zbl

[15] Lempert, László La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. Fr., Volume 109 (1981) no. 4, pp. 427-474 | DOI | MR | Zbl

[16] Lempert, László; Vivas, Liz Geodesics in the space of Kähler metrics, Duke Math. J., Volume 162 (2013) no. 7, pp. 1369-1381 | DOI | MR | Zbl

[17] Mabuchi, Toshiki Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987) no. 2, pp. 227-252 http://projecteuclid.org/euclid.ojm/1200780161 | MR | Zbl

[18] Richardson, Stanley Hele-Shaw flows with a free boundary produced by the injection of a fluid into a narrow channel, J. Fluid Mech., Volume 56 (1972) no. 4, p. 609-18 | Zbl

[19] Ross, Julius; Nyström, David Witt Harmonic discs of solutions to the complex homogeneous Monge–Ampère equation, Publ. Math., Inst. Hautes Étud. Sci., Volume 122 (2015), pp. 315-335 | DOI | MR | Zbl

[20] Ross, Julius; Nyström, David Witt The Hele-Shaw flow and moduli of holomorphic discs, Compos. Math., Volume 151 (2015) no. 12, pp. 2301-2328 | DOI | MR | Zbl

[21] Ross, Julius; Nyström, David Witt Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry., Int. Math. Res. Not. (2017) no. 23, pp. 7069-7108 | Zbl

[22] Rubinstein, Yanir A.; Zelditch, Steve The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform, Adv. Math., Volume 228 (2011) no. 6, pp. 2989-3025 | DOI | MR | Zbl

[23] Rubinstein, Yanir A.; Zelditch, Steve The Cauchy problem for the homogeneous Monge–Ampère equation, I. Toeplitz quantization, J. Differ. Geom., Volume 90 (2012) no. 2, pp. 303-327 http://projecteuclid.org/euclid.jdg/1335230849 | DOI | MR | Zbl

[24] Rubinstein, Yanir A.; Zelditch, Steve The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan, J. Reine Angew. Math., Volume 724 (2017), pp. 105-143 | DOI | MR | Zbl

[25] Semmes, Stephen Complex Monge–Ampère and symplectic manifolds, Am. J. Math., Volume 114 (1992) no. 3, pp. 495-550 | DOI | MR | Zbl

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