We study an equation proposed by Fu and Yau as a natural n-dimensional generalization of a Strominger system that they solved in dimension 2. It is a complex Hessian equation with right-hand side depending on gradients. Building on the methods of Fu and Yau, we obtain , , and a priori estimates. We also identify difficulties in extending the Fu–Yau arguments for non-degeneracy from dimension 2 to higher dimensions.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-12-66033
Award Identifier / Grant number: DMS-1308136
Funding statement: Work supported in part by the National Science Foundation under Grant DMS-12-66033 and DMS-1308136.
The authors would like to thank Pengfei Guan for stimulating conversations and for his notes on Fu–Yau’s equation. They would also like to thank Valentino Tosatti for his lectures and notes on Strominger systems. The authors are also very grateful to the referee for a particularly careful reading of the paper, and for numerous suggestions which helped clarify the paper a great deal.
 S. Dinew and S. Kolodziej, Liouville and Calabi–Yau type theorems for complex Hessian equations, preprint (2012), https://arxiv.org/abs/1203.3995; to appear in Amer. J. Math. 10.1353/ajm.2017.0009Search in Google Scholar
 J.-X. Fu and S. T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation, J. Differential Geom. 78 (2008), no. 3, 369–428. 10.4310/jdg/1207834550Search in Google Scholar
 P. Guan, On the gradient estimate for Monge–Ampère equation on Kähler manifold, private notes. Search in Google Scholar
 Z. Hou, X.-N. Ma and D. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett. 17 (2010), no. 3, 547–561. 10.4310/MRL.2010.v17.n3.a12Search in Google Scholar
 S. Kolodziej and V.-D. Nguyen, Weak solutions of complex Hessian equations on compact Hermitian manifolds, preprint (2015), https://arxiv.org/abs/1507.06755. 10.1112/S0010437X16007417Search in Google Scholar
 N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations (Russian), Izv. Akad. Nauk. SSSR 46 (1982), 487–523; translation in Math. USSR Izv. 20 (1983), no. 3, 459–492. 10.1070/IM1983v020n03ABEH001360Search in Google Scholar
 D. H. Phong, J. Song and J. Sturm, Complex Monge–Ampère equations, In memory of C. C. Hsiung. Lectures given at the JDG symposium on geometry and topology (Bethlehem, PA, 2010), Surv. Differ. Geom. 17, International Press, Somerville (2012), 327–410. 10.4310/SDG.2012.v17.n1.a8Search in Google Scholar
 V. Tosatti, Y. Wang, B. Weinkove and X. Yang, estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 431–453. 10.1007/s00526-014-0791-0Search in Google Scholar
 V. Tosatti and B. Weinkove, The complex Monge–Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. 10.1090/S0894-0347-2010-00673-XSearch in Google Scholar
 Y. Wang, On the regularity of the complex Monge–Ampère equation, Math. Res. Lett. 19 (2012), no. 4, 939–946. Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston