Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T19:03:36.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak solutions of complex Hessian equations on compact Hermitian manifolds

Part of: Global differential geometry Elliptic equations and systems Pluripotential theory

Published online by Cambridge University Press:  09 September 2016

Sławomir Kołodziej  and
Ngoc Cuong Nguyen
Sławomir Kołodziej
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Łojasiewicza 6, Poland email [email protected]
Ngoc Cuong Nguyen
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Łojasiewicza 6, Poland email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

Keywords

weak solutionscomplex Hessian equationcomparison principle

MSC classification

Primary: 53C55: Hermitian and Kählerian manifolds 35J96: Elliptic Monge-Ampère equations 32U40: Currents
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2221 - 2248
Copyright
© The Authors 2016 

References

Alekser, S. and Verbitsky, M., Quaternionic Monge–Ampère equations and Calabi problem for HKT-manifolds , Israel J. Math. 176 (2010), 109138.Google Scholar
Bedford, E. and Taylor, B. A., The Dirichlet problem for a complex Monge–Ampère operator , Invent. Math. 37 (1976), 1–44.CrossRefGoogle Scholar
Bedford, E. and Taylor, B. A., A new capacity for plurisubharmonic functions , Acta Math. 149 (1982), 1–40.CrossRefGoogle Scholar
Berman, R., From Monge–Ampere equations to envelopes and geodesic rays in the zero temperature limit. Preprint (2013), arXiv:1307.3008.Google Scholar
Błocki, Z., Weak solutions to the complex Hessian equation , Ann. Inst. Fourier (Grenoble) 55 (2005), 1735–1756.CrossRefGoogle Scholar
Błocki, Z. and Kołodziej, S., On regularization of plurisubharmonic functions on manifolds , Proc. Amer. Math. Soc. 135 (2007), 2089–2093.CrossRefGoogle Scholar
Caffarelli, L., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian , Acta Math. 155 (1985), 261–301.CrossRefGoogle Scholar
Demailly, J.-P. and Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold , Ann. of Math. (2) 159 (2004), 1247–1274.CrossRefGoogle Scholar
Dinew, S. and Kołodziej, S., Pluripotential estimates on compact Hermitian manifolds , in Advances in geometric analysis, Advanced Lectures in Mathematics, vol. 21 (International Press, Boston, 2012).Google Scholar
Dinew, S. and Kołodziej, S., Liouville and Calabi–Yau type theorems for complex Hessian equations. Preprint (2012), arXiv:1203.3995, Amer. J. Math., to appear.Google Scholar
Dinew, S. and Kołodziej, S., A priori estimates for complex Hessian equations , Anal. PDE 7 (2014), 227–244.CrossRefGoogle Scholar
Dinew, S. and Lu, C. H., Mixed Hessian inequalities and uniqueness in the class E(X, 𝜔, m) , Math. Z. 279 (2015), 753–766.CrossRefGoogle Scholar
Eyssidieux, P., Guedj, V. and Zeriahi, A., Continuous approximation of quasiplurisubharmonic functions , in Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemporary Mathematics, vol. 644 (American Mathematical Society, Providence, RI, 2015), 67–78; MR3372461.CrossRefGoogle Scholar
Fu, J.-X. and Yau, S.-T., The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation , J. Differential Geom. 98 (2008), 369–428.Google Scholar
Gårding, L., An inequality for hyperbolic polynomials , J. Math. Mech. 8 (1959), 957–965.Google Scholar
Hörmander, L., Notions of convexity, Progress in Mathematics, vol. 127 (Birkhäuser, Boston, 1994).Google Scholar
Hou, Z., Ma, X.-N. and Wu, D., A second order estimate for complex Hessian equations on a compact Kähler manifold , Math. Res. Lett. 17 (2010), 547–561.CrossRefGoogle Scholar
Ivochkina, N. M., Description of cones of stability generated by differential operators of Monge–Ampère type , Mat. Sb. (N.S.) 122 (1983), 265–275.Google Scholar
Kołodziej, S., The complex Monge–Ampère equation , Acta Math. 180 (1998), 69–117.CrossRefGoogle Scholar
Kołodziej, S., The Monge–Ampère equation on compact Kähler manifolds , Indiana Univ. Math. J. 52 (2003), 667–686.CrossRefGoogle Scholar
Kołodziej, S., The complex Monge–Ampère equation and pluripotential theory , Mem. Amer. Math. Soc. 178 (2005), 64.Google Scholar
Kołodziej, S. and Nguyen, N.-C., Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds , in Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemporary Mathematics, vol. 644 (American Mathematical Society, Providence, RI, 2015), 141–158.CrossRefGoogle Scholar
Kołodziej, S. and Nguyen, N.-C., Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifold. Preprint (2015), arXiv:1501.05749.Google Scholar
Lin, M. and Trudinger, N. S., On some inequalities for elementary symmetric functions , Bull. Aust. Math. Soc. 50 (1994), 317–326.CrossRefGoogle Scholar
Lu, H. C., Solutions to degenerate complex Hessian equations , J. Math. Pures Appl. (9) 100 (2013), 785–805.CrossRefGoogle Scholar
Lu, C. H. and Nguyen, Van-Dong, Degenerate complex Hessian equations on compact Kähler manifolds , Indiana Univ. Math. J. 64 (2015), 1721–1745; MR3436233.CrossRefGoogle Scholar
Michelsohn, M. L., On the existence of special metrics in complex geometry , Acta Math. 149 (1982), 261–295.CrossRefGoogle Scholar
Nguyen, N.-C., The complex Monge–Ampère type equation on compact Hermitian manifolds and Applications , Adv. Math. 286 (2016), 240–285; MR3415685.CrossRefGoogle Scholar
Phong, D. H., Picard, S. and Zhang, X., On estimates for the Fu–Yau generalization of a Strominger system. Preprint (2015), arXiv:1507.08193.Google Scholar
Székelyhidi, G., Fully non-linear elliptic equations on compact Hermitian manifolds , J. Differential Geom., to appear. Preprint (2015), arXiv:1501.02762v3.Google Scholar
Székelyhidi, G., Tosatti, V. and Weinkove, B., Gauduchon metrics with prescribed volume form. Preprint (2015), arXiv:1503.04491.Google Scholar
Tosatti, V., Wang, Y., Weinkove, B. and Yang, X., C 2, 𝛼 estimates for nonlinear elliptic equations in complex and almost complex geometry , Calc. Var. Partial Differential Equations 54 (2015), 431–453; MR3385166.CrossRefGoogle Scholar
Tosatti, V. and Weinkove, B., The complex Monge–Ampère equation on compact Hermitian manifolds , J. Amer. Math. Soc. 23 (2010), 1187–1195.CrossRefGoogle Scholar
Tosatti, V. and Weinkove, B., Hermitian metrics, $(n-1,n-1)$ forms and Monge–Ampère equations. Preprint (2013), arXiv:1310.6326.Google Scholar
Wang, X.-J., The k-Hessian equation , in Geometric analysis and PDEs, Lecture Notes in Mathematics, vol. 1977 (Springer, Dordrecht, 2009), 177–252.CrossRefGoogle Scholar
Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation , Comm. Pure Appl. Math. 31 (1978), 339–411.CrossRefGoogle Scholar
Zhang, D., Hessian equations on closed Hermitian manifolds. Preprint (2015),arXiv:1501.03553.Google Scholar