Skip to main content
Log in

\(C^{1,1}\) regularity of geodesics in the space of volume forms

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove a \(C^{1,1}\) estimate for solutions of a class of fully nonlinear equations introduced by Chen–He. As an application, we prove the \(C^{1,1}\) regularity of geodesics in the space of volume forms.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Błocki, Z.: Regularity of the degenerate Monge–Ampère equation on compact Kähler manifolds. Math. Z. 244, 153–161 (2003)

    Article  MathSciNet  Google Scholar 

  2. Chen, X.X., He, W.Y.: The space of volume forms. Int. Math. Res. Not. IMRN 5, 967–1009 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Chen, X.X., He, W.Y.: A class of fully nonlinear equations. Preprint arXiv:1802.04985

  4. Chu, J., Tosatti, V., Weinkove, B.: On the \(C^{1,1}\) regularity of geodesics in the space of Kähler metrics. Ann. PDE 3(2), Art. 15, (2017)

  5. Darvas, T.: Morse theory and geodesics in the space of Kähler metrics. Proc. Am. Math. Soc. 142(8), 2775–2782 (2014)

    Article  Google Scholar 

  6. Darvas, T., Lempert, L.: Weak geodesics in the space of Kähler metrics. Math. Res. Lett. 19(5), 1127–1135 (2012)

    Article  MathSciNet  Google Scholar 

  7. Donaldson, S.: Nahm’s Equations and Free-Boundary Problems, The Many Facets of Geometry, pp. 71–91. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  8. Gursky, M., Streets, J.: A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma _{2}\)-Yamabe problem. Geom. Topol. 22(6), 3501–3573 (2018)

    Article  MathSciNet  Google Scholar 

  9. He, W.Y.: The Donaldson equation. Preprint arXiv:0810.4123

  10. Lempert, L., Vivas, L.: Geodesics in the space of Kähler metrics. Duke Math. J. 162(7), 1369–1381 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jianchun Chu.

Additional information

Communicated by A.Chang.

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

Chu, J. \(C^{1,1}\) regularity of geodesics in the space of volume forms. Calc. Var. 58, 194 (2019). https://doi.org/10.1007/s00526-019-1648-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1648-3

Mathematics Subject Classification

Navigation