Skip to main content
SpringerLink
Log in

A variational approach to the quaternionic Monge–Ampère equation

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

In this paper, we use the variational method to solve the quaternionic Monge–Ampère equation when the right-hand side is a positive measure of finite energy. We introduce finite energy classes of quaternionic plurisubharmonic functions of Cegrell type and define the quaternionic Monge–Ampère operator on some Cegrell’s classes, the functions of which are not necessarily bounded. By using the theory of quaternionic closed positive current, we show that integration by parts and comparison principle are valid on some classes. This opens the door to prove results in the quaternionic pluripotential theory as in the seminal framework by Cegrell (Acta Math 180(2):187–217, 1998; Ann Inst Fourier (Grenoble) 54(1):159–179, 2004; Ann Polon Math 94(2):131–147, 2008) for the complex Monge–Ampère case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Åhag, P., Cegrell, U., Kołodziej, S., Phạm, H.H., Zeriahi, A.: Partial pluricomplex energy and integrability exponents of plurisubharmonic functions. Adv. Math. 222(6), 2036–2058 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Åhag, P., Cegrell, U., Czyż, R.: On Dirichlet’s principle and problem. Math. Scand. 110(2), 235–250 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alesker, S.: Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables. Bull. Sci. Math. 127(1), 1–35 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alesker, S.: Quaternionic Monge–Ampère equations. J. Geom. Anal. 13(2), 205–238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alesker, S.: Pluripotential theory on quaternionic manifolds. J. Geom. Phys. 62(5), 1189–1206 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alesker, S.: Solvability of the quaternionic Monge–Ampère equation on compact manifolds with a flat hyperKähler metric. Adv. Math. 241, 192–219 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alesker, S., Shelukhin, E.: On a uniform estimate for the quaternionic Calabi problem. Isr. J. Math. 197(1), 309–327 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alesker, S., Verbitsky, M.: Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry. J. Geom. Anal. 16(3), 375–399 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Alesker, S., Verbitsky, M.: Quaternionic Monge–Ampère equation and Calabi problem for HKT-manifolds. Isr. J. Math. 176, 109–138 (2010)

    Article  MATH  Google Scholar 

  10. Bedford, E.: Survey of pluri-potential theory. In: Several complex variables (Stockholm, 1987/1988), volume 38 of Math. Notes, pp. 48–97. Princeton University Press, Princeton, NJ 1(993)

  11. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37(1), 1–44 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2), 1–40 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Berman, R., Boucksom, S., Nyström, D.W.: Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207(1), 1–27 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Berman, R.J., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampère equations. Publ. Math. Inst. Hautes Études Sci. 117, 179–245 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cegrell, U.: Approximation of plurisubharmonic functions in hyperconvex domains. In: Complex Analysis and Digital Geometry, volume 86 of Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., pp. 125–129. Uppsala Universitet, Uppsala (2009)

  16. Cegrell, U.: Sums of continuous plurisubharmonic functions and the complex Monge–Ampère operator in \({ C}^n\). Math. Z. 193(3), 373–380 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cegrell, U.: Pluricomplex energy. Acta Math. 180(2), 187–217 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble) 54(1), 159–179 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cegrell, U.: A general Dirichlet problem for the complex Monge-Ampère operator. Ann. Polon. Math. 94(2), 131–147 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Demailly, J.P.: Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France (N.S.) 19, 124 (1985)

    MATH  Google Scholar 

  21. Demailly, J.-P., Pham, H.H.: A sharp lower bound for the log canonical threshold. Acta Math. 212(1), 1–9 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kang, Q., Wang, W.: On Penrose integral formula and series expansion of \(k\)-regular functions on the quaternionic space \({\mathbb{H}}^n\). J. Geom. Phys. 64, 192–208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kiselman, C.O.: Sur la définition de l’opérateur de Monge-Ampère complexe. In: Complex Analysis (Toulouse, 1983), volume 1094 of Lecture Notes in Math., pp. 139–150. Springer, Berlin (1984)

  24. Klimek, M.: Pluripotential theory, volume 6 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1991)

  25. Lu, C.H.: A variational approach to complex Hessian equations in \({\mathbb{C}}^n\). J. Math. Anal. Appl. 431(1), 228–259 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Persson, L.: A Dirichlet principle for the complex Monge–Ampère operator. Ark. Mat. 37(2), 345–356 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sroka, M.: Weak solutions to the quaternionic Monge–Ampère equation. arXiv:1807.02482

  28. Van Khue, N., Hiep, P.H.: A comparison principle for the complex Monge–Ampère operator in Cegrell’s classes and applications. Trans. Am. Math. Soc. 361(10), 5539–5554 (2009)

    Article  MATH  Google Scholar 

  29. Wan, D.: The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space. Math. Z. 285, 461–478 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wan, D., Kang, Q.: Potential theory for quaternionic plurisubharmonic functions. Mich. Math. J. 66, 3–20 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wan, D., Wang, W.: Viscosity solutions to quaternionic Monge–Ampère equations. Nonlinear Anal. 140, 69–81 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wan, D., Wang, W.: On quaternionic Monge–Ampère operator, closed positive currents and Lelong–Jensen type formula on quaternionic space. Bull. Sci. Math. 141, 267–311 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wan, D., Zhang, W.: Quasicontinuity and maximality of quaternionic plurisubharmonic functions. J. Math. Anal. Appl. 424, 86–103 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, W.: On non-homogeneous Cauchy–Fueter equations and Hartogs’ phenomenon in several quaternionic variables. J. Geom. Phys. 58(9), 1203–1210 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, W.: The \(k\)-Cauchy–Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic \(k\)-regular functions. J. Geom. Phys. 60(3), 513–530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, W.: On the optimal control method in quaternionic analysis. Bull. Sci. Math. 135(8), 988–1010 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, W.: The tangential Cauchy–Fueter complex on the quaternionic Heisenberg group. J. Geom. Phys. 61(1), 363–380 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, W.: On quaternionic complexes over unimodular quaternionic manifolds. Differ. Geom. Appl. 58, 227–253 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Nature Science Foundation in China (No. 11871345).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Room 312 Science and Technology Building, Shenzhen, 518060, People’s Republic of China

    Dongrui Wan

Authors
  1. Dongrui Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongrui Wan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wan, D. A variational approach to the quaternionic Monge–Ampère equation. Annali di Matematica 199, 2125–2150 (2020). https://doi.org/10.1007/s10231-020-00960-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-020-00960-z

Keywords

Mathematics Subject Classifcation

Search

Navigation