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St. Petersburg Mathematical Journal

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Quaternionic plurisubharmonic functions and their applications to convexity


Author: S. Alesker
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 1-13
MSC (2000): Primary 31C10, 52A38, 52A39
DOI: https://doi.org/10.1090/S1061-0022-07-00982-X
Published electronically: December 12, 2007
MathSciNet review: 2319507
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Abstract | References | Similar Articles | Additional Information

Abstract: 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 some earlier papers by the author and a joint paper by Verbitsky and the author.


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  • 1. A. D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern. IV. Die gemischten Diskriminanten und die gemischten Volumina, Mat. Sb. (N.S.) 3 (45) (1938), no. 2, 227-251. (Russian) (German summary).
  • 2. -, Dirichlet's problem for the equation $ \operatorname{Det}\Vert z_{ij}\Vert =\varphi (z_{1},\dots,z_{n},z, x_{1},\dots, x_{n})$. I, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1958, vyp. 1, 5-24. (Russian) MR 0096903 (20:3385)
  • 3. S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244-272. MR 1837364 (2002e:52015)
  • 4. -, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, Bull. Sci. Math. 127 (2003), no. 1, 1-35; also: math.CV/0104209. MR 1957796 (2004b:32054)
  • 5. -, Quaternionic Monge-Ampère equations, J. Geom. Anal. 13 (2003), no. 2, 205-238; also: math.CV/0208005. MR 1967025 (2004d:32055)
  • 6. -, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (2003), 63-95; also: math.MG/0209263. MR 2015260 (2004h:52015)
  • 7. -, Valuations on convex sets, non-commutative determinants, and pluripotential theory, Adv. Math. 195 (2005), no. 2, 561-595; also: math.MG/0401219. MR 2146354 (2006f:32047)
  • 8. S. Alesker and M. Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, J. Geom. Anal. 16 (2006), no. 3, 375-399; also: math.CV/0510140. MR 2250051 (2007e:32042)
  • 9. E. Artin, Geometric algebra, Intersci. Publ., Inc., New York-London, 1957. MR 0082463 (18:553e)
  • 10. H. Aslaksen, Quaternionic determinants, Math. Intelligencer 18 (1996), no. 3, 57-65. MR 1412993 (97j:16028)
  • 11. B. Banos and A. Swann, Potentials for hyper-Kähler metrics with torsion, Classical Quantum Gravity 21 (2004), no. 13, 3127-3135. MR 2072130 (2005c:53054)
  • 12. E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, 1-44. MR 0445006 (56:3351)
  • 13. Z. B\locki, Equilibrium measure of a product subset of $ C^n$, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3595-3599. MR 1707508 (2001b:32072)
  • 14. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, ``Nauka'', Leningrad, 1980; English transl., Grundlehren Math. Wiss., Bd. 285, Springer-Verlag, Berlin, 1988. MR 0602952 (82d:52009); MR 0936419 (89b:52020)
  • 15. L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369-402. MR 0739925 (87f:35096)
  • 16. L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38 (1985), no. 2, 209-252. MR 0780073 (87f:35097)
  • 17. Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $ \det(\partial \sp{2}u/\partial x\sb{i}\partial x\sb{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), no. 1, 41-68. MR 0437805 (55:10727)
  • 18. -, The real Monge-Ampère equation and affine flat structures, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vols. 1, 2, 3 (Beijing, 1980), Science Press, Beijing, 1982, pp. 339-370. MR 0714338 (85c:53103)
  • 19. S. S. Chern, H. I. Levine, and L. Nirenberg, Intrinsic norms on a complex manifold, 1969 Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 119-139. MR 0254877 (40:8084)
  • 20. I. Gelfand, S. Gelfand, V. Retakh, and R. Lee Wilson, Quasideterminants, Adv. Math. 193 (2005), no. 1, 56-141. MR 2132761 (2006a:05165)
  • 21. I. Gelfand, V. Retakh, and R. Lee Wilson, Quaternionic quasideterminants and determinants, Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 111-123; also: math.QA/0206211. MR 2018356 (2004j:15010)
  • 22. G. Grantcharov and Y. S. Poon, Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys. 213 (2000), no. 1, 19-37. MR 1782143 (2002a:53059)
  • 23. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957. MR 0102775 (21:1561)
  • 24. G. Henkin, Private communication.
  • 25. P. S. Howe and G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B 379 (1996), no. 1-4, 80-86. MR 1396267 (97h:53073)
  • 26. B. Ya. Kazarnovskii, On zeros of exponential sums, Dokl. Akad. Nauk SSSR 257 (1981), no. 4, 804-808; English transl. in Soviet Math. Dokl. 23 (1981). MR 0612571 (82i:32014)
  • 27. -, Newton polyhedra and roots of systems of exponential sums, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 40-49; English transl., Funct. Anal. Appl. 18 (1984), no. 4, 299-307. MR 0775932 (87b:32005)
  • 28. D. Klain and G.-C. Rota, Introduction to geometric probability, Lincei Lectures, Cambridge Univ. Press, Cambridge, 1997. MR 1608265 (2001f:52009)
  • 29. N. V. Krylov, Smoothness of the payoff function for a controllable diffusion process in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 66-96; English transl., Math. USSR-Iz. 34 (1990), no. 1, 65-95. MR 0992979 (90f:93040)
  • 30. P. McMullen, Continuous translation-invariant valuations on the space of compact convex sets, Arch. Math. (Basel) 34 (1980), no. 4, 377-384. MR 0593954 (81m:52013)
  • 31. -, Valuations and dissections, Handbook of Convex Geometry, Vols. A, B, North-Holland, Amsterdam, 1993, pp. 933-988. MR 1243000 (95f:52018)
  • 32. P. McMullen and R. Schneider, Valuations on convex bodies, Convexity and its Applications, Birkhäuser, Basel, 1983, pp. 170-247. MR 0731112 (85e:52001)
  • 33. E. H. Moore, On the determinant of an hermitian matrix of quaternionic elements, Bull. Amer. Math. Soc. 28 (1922), 161-162.
  • 34. A. V. Pogorelov, The regularity of the generalized solutions of the equation $ \det(\partial \sp{2}u/\partial x\sp{i}\partial x\sp{j})=\varphi (x\sp{1},\,x\sp{2},\dots, x\sp{n})>0$, Dokl. Akad. Nauk SSSR 200 (1971), no. 3, 534-537; English transl., Soviet Math. Dokl. 12 (1971), 1436-1440. MR 0293227 (45:2304)
  • 35. -, The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation, Dokl. Akad. Nauk SSSR 201 (1971), no. 4, 790-793; English transl., Soviet Math. Dokl. 12 (1971), 1227-1231. MR 0293228 (45:2305)
  • 36. -, A regular solution of the $ n$-dimensional Minkowski problem, Dokl. Akad. Nauk SSSR 199 (1971), no. 4, 785-788; English transl., Soviet Math. Dokl. 12 (1971), 1192-1196. MR 0284956 (44:2180)
  • 37. -, The multidimensional Monge-Ampère equation $ {\rm det}\,\Vert z\sb {ij}\Vert =\phi(z\sb 1,\dots,z\sb n,z, x\sb 1,\dots,x\sb n)$, ``Nauka'', Moscow, 1988. (Russian) MR 0938951 (89g:35036)
  • 38. R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • 39. M. Verbitsky, Hyper-Kähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), no. 4, 679-712. MR 1958088 (2004e:53071)

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Additional Information

S. Alesker
Affiliation: Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
Email: semyon@post.tau.ac.il

DOI: https://doi.org/10.1090/S1061-0022-07-00982-X
Keywords: HKT-geometry, valuation on convex sets, quaternionic plurisubharmonic functions
Received by editor(s): August 1, 2006
Published electronically: December 12, 2007
Additional Notes: Partially supported by ISF (grant 1369/04)
Dedicated: Dedicated to Professor Victor Abramovich Zalgaller on the occasion of his 85th birthday
Article copyright: © Copyright 2007 American Mathematical Society

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