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The parabolic Monge–Ampère equation on compact almost Hermitian manifolds

  • Jianchun Chu
Published/Copyright: August 7, 2018
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 761

Abstract

We prove the long time existence and uniqueness of solutions to the parabolic Monge–Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in C∞ topology as t→∞. Up to scaling, the limit function is a solution of the Monge–Ampère equation. This gives a parabolic proof of existence of solutions to the Monge–Ampère equation on almost Hermitian manifolds.

Acknowledgements

The author would like to thank his advisor Professor Gang Tian for encouragement and support, and Professor Valentino Tosatti and Professor Ben Weinkove for suggesting this problem and helpful suggestions. This work was carried out while the author was visiting the Department of Mathematics at Northwestern University, supported by the China Scholarship Council (File No. 201506010010). The author would like to thank the China Scholarship Council for supporting this visiting, and the Department of Mathematics at Northwestern University for its hospitality and for providing a good academic environment.

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Received: 2016-08-29
Published Online: 2018-08-07
Published in Print: 2020-04-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The parabolic Monge–Ampère equation on compact almost Hermitian manifolds
  3. Gaussian heat kernel estimates: From functions to forms
  4. Flat parabolic vector bundles on elliptic curves
  5. A Zariski--Nagata theorem for smooth ℤ-algebras
  6. Ordinary K3 surfaces over a finite field
  7. The uniformization of the moduli space of principally polarized abelian 6-folds
  8. Wahl maps and extensions of canonical curves and K⁢3 surfaces
  9. A new proof of Kirchberg's 𝒪2-stable classification
  10. Erratum to On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (J. reine angew. Math. 713 (2016), 21–48)
https://doi.org/10.1515/crelle-2018-0019

Articles in the same Issue

  1. Frontmatter
  2. The parabolic Monge–Ampère equation on compact almost Hermitian manifolds
  3. Gaussian heat kernel estimates: From functions to forms
  4. Flat parabolic vector bundles on elliptic curves
  5. A Zariski--Nagata theorem for smooth ℤ-algebras
  6. Ordinary K3 surfaces over a finite field
  7. The uniformization of the moduli space of principally polarized abelian 6-folds
  8. Wahl maps and extensions of canonical curves and K⁢3 surfaces
  9. A new proof of Kirchberg's 𝒪2-stable classification
  10. Erratum to On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (J. reine angew. Math. 713 (2016), 21–48)
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