Quaternionic plurisubharmonic functions and their applications to convexity
Algebra i analiz, Tome 19 (2007) no. 1, pp. 5-22.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is a survey of the recent theory of plurisubharmonic functions of quaternionic variables, together with its applications to the theory of valuations on convex sets and HKT-geometry (Hyper-Kähler with Torsion). The exposition follows the papers [4, 5, 7] by the author and [8] by Verbitsky and the author.
Keywords: HKT-geometry, valuation on convex sets, quaternionic plurisubharmonic functions.
@article{AA_2007_19_1_a0,
     author = {S. Alesker},
     title = {Quaternionic plurisubharmonic functions and their applications to convexity},
     journal = {Algebra i analiz},
     pages = {5--22},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2007_19_1_a0/}
}
TY  - JOUR
AU  - S. Alesker
TI  - Quaternionic plurisubharmonic functions and their applications to convexity
JO  - Algebra i analiz
PY  - 2007
SP  - 5
EP  - 22
VL  - 19
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2007_19_1_a0/
LA  - en
ID  - AA_2007_19_1_a0
ER  - 
%0 Journal Article
%A S. Alesker
%T Quaternionic plurisubharmonic functions and their applications to convexity
%J Algebra i analiz
%D 2007
%P 5-22
%V 19
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2007_19_1_a0/
%G en
%F AA_2007_19_1_a0
S. Alesker. Quaternionic plurisubharmonic functions and their applications to convexity. Algebra i analiz, Tome 19 (2007) no. 1, pp. 5-22. http://geodesic.mathdoc.fr/item/AA_2007_19_1_a0/

[1] Aleksandrov A. D., “K teorii smeshannykh ob'emov vypuklykh tel. IV. Smeshannye diskriminanty i smeshannye ob'emy”, Matem. sb., 3(45):2 (1938), 227–251 | Zbl

[2] Aleksandrov A. D., “Zadacha Dirikhle dlya uravneniya $\operatorname{Det}||z_{ij}||=\varphi(z_1,\dots,z_n,z,x_1,\dots,x_n)$”, Vestn. Leningr. un-ta. Ser. mat., mekh., astronom., 1958, no. 1, 5–24 | Zbl

[3] Alesker S., “Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture”, Geom. Funct. Anal., 11:2 (2001), 244–272 | DOI | MR | Zbl

[4] Alesker S., “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables”, Bull. Sci. Math., 127:1 (2003), 1–35 ; also: arXiv: math.CV/0104209 | DOI | MR | Zbl

[5] Alesker S., “Quaternionic Monge–Ampère equations”, J. Geom. Anal., 13:2 (2003), 205–238 ; also: arXiv: math.CV/0208005 | MR

[6] Alesker S., “Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations”, J. Differential Geom., 63:1 (2003), 63–95 ; also: arXiv: math.MG/0209263 | MR | Zbl

[7] Alesker S., “Valuations on convex sets, non-commutative determinants, and pluripotential theory”, Adv. Math., 195:2 (2005), 561–595 ; also: arXiv: math.MG/0401219 | DOI | MR | Zbl

[8] Alesker S., Verbitsky M., “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry”, J. Geom. Anal., 16:3 (2006), 375–399 ; also: arXiv: math.CV/0510140 | MR | Zbl

[9] Artin E., Geometric algebra, Intersci. Publ., Inc., New York–London, 1957 | MR | Zbl

[10] Aslaksen H., “Quaternionic determinants”, Math. Intelligencer, 18:3 (1996), 57–65 | DOI | MR | Zbl

[11] Banos B., Swann A., “Potentials for hyper-Kähler metrics with torsion”, Classical Quantum Gravity, 21:13 (2004), 3127–3135 | DOI | MR | Zbl

[12] Bedford E., Taylor B. A., “The Dirichlet problem for a complex Monge–Ampère equation”, Invent. Math., 37:1 (1976), 1–44 | DOI | MR | Zbl

[13] Błocki Z., “Equilibrium measure of a product subset of $C^n$”, Proc. Amer. Math. Soc., 128:12 (2000), 3595–3599 | DOI | MR

[14] Burago Yu. V., Zalgaller V. A., Geometricheskie neravenstva, Nauka, L., 1980 | MR | Zbl

[15] Caffarelli L., Nirenberg L., Spruck J., “The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge–Ampère equation”, Comm. Pure Appl. Math., 37:3 (1984), 369–402 | DOI | MR | Zbl

[16] Caffarelli L., Kohn J. J., Nirenberg L., Spruck J., “The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge–Ampère, and uniformly elliptic quations”, Comm. Pure Appl. Math., 38:2 (1985), 209–252 | DOI | MR | Zbl

[17] Cheng Shiu Yuen, Yau Shing Tung, “On the regularity of the Monge–Amp`ere equation $\det(\partial^2u\partial x_i\partial s x_j)=F(x,u)$”, Comm. Pure Appl. Math., 30:1 (1977), 41–68 | DOI | MR | Zbl

[18] Cheng Shiu Yuen, Yau Shing-Tung, “The real Monge–Ampère equation and affine flat structures”, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), Science Press, Beijing, 1982, 339–370 | MR

[19] Chern S. S., Levine H. I., Nirenberg L., “Intrinsic norms on a complex manifold”, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, 119–139 | MR

[20] Gelfand I., Gelfand S., Retakh V., Wilson R. Lee, “Quasideterminants”, Adv. Math., 193:1 (2005), 56–141 | DOI | MR | Zbl

[21] Gelfand I., Retakh V., Wilson R. Lee, “Quaternionic quasideterminants and determinants, Lie Groups and Symmetric Spaces”, Amer. Math. Soc. Transl. Ser. 2, 210, AMS, Providence, RI, 2003, 111–123 ; also: arXiv: math.QA/0206211 | MR | Zbl

[22] Grantcharov G., Poon Y. S., “Geometry of hyper-Kähler connections with torsion”, Comm. Math. Phys., 213:1 (2000), 19–37 | DOI | MR | Zbl

[23] Hadwiger H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin etc., 1957 | MR

[24] Henkin G., Private communication

[25] Howe P. S., Papadopoulos G., “Twistor spaces for hyper-Kähler manifolds with torsion”, Phys. Lett. B, 379:1–4 (1996), 80–86 | MR

[26] Kazarnovskii B. Ya., “O nulyakh eksponentsialnykh summ”, Dokl. AN SSSR, 257:4 (1981), 804–808 | MR | Zbl

[27] Kazarnovskii B. Ya., “Mnogogranniki Nyutona i korni sistem eksponentsialnykh summ”, Funkts. analiz i ego pril., 18:4 (1984), 40–49 | MR | Zbl

[28] Klain D., Rota G.-C., Introduction to geometric probability, Lincei Lectures, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[29] Krylov N. V., “Gladkost funktsii vyigrysha dlya upravlyaemogo diffuzionnogo protsessa v oblasti”, Izv. AN SSSR. Ser. matem., 53:1 (1989), 66–96 | MR

[30] McMullen P., “Continuous translation-invariant valuations on the space of compact convex sets”, Arch. Math. (Basel), 34:4 (1980), 377–384 | DOI | MR | Zbl

[31] McMullen P., “Valuations and dissections”, Handbook of Convex Geometry, Vol. A, B, North-Holland, Amsterdam, 1993, 933–988 | MR | Zbl

[32] McMullen P., Schneider R., “Valuations on convex bodies”, Convexity and its Applications, Birkhäuser, Basel, 1983, 170–247 | MR

[33] Moore E. H., “On the determinant of an hermitian matrix of quaternionic elements”, Bull. Amer. Math. Soc., 28 (1922), 161–162

[34] Pogorelov A. V., “O regulyarnosti obobschennykh reshenii uravneniya $\det(\partial^2u\partial x^i\partial s x^j)=\varphi(x^1,x^2,\dots,x^n)>0$”, Dokl. AN SSSR, 200:3 (1971), 534–537 | MR | Zbl

[35] Pogorelov A. V., “Zadacha Dirikhle dlya mnogomernogo analoga uravneniya Monzha–Ampera”, Dokl. AN SSSR, 201:4 (1971), 790–793 | MR | Zbl

[36] Pogorelov A. V., “Regulyarnoe reshenie $n$-mernoi problemy Minkovskogo”, Dokl. AN SSSR, 199:4 (1971), 785–788 | MR | Zbl

[37] Pogorelov A. V., Mnogomernoe uravnenie Monzha–Ampera $\det\|z_{ij}\|=\phi(z_1,\dots,z_n,z,x_1,\dots,x_n)$, Nauka, M., 1988 | MR

[38] Schneider R., Convex bodies: the Brunn–Minkowski theory, Encyclopedia Math. Appl., 44, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[39] Verbitsky M., “Hyper-Kähler manifolds with torsion, supersymmetry and Hodge theory”, Asian J. Math., 6:4 (2002), 679–712 | MR | Zbl