A type of parabolic flow with mixed hessians on compact Kähler manifolds

Tsai, Yi-Lin

doi:10.1090/proc/16097

A type of parabolic flow with mixed hessians on compact Kähler manifolds
HTML articles powered by AMS MathViewer

by Yi-Lin Tsai;

Proc. Amer. Math. Soc. 151 (2023), 425-437

DOI: https://doi.org/10.1090/proc/16097

Published electronically: August 19, 2022

HTML | PDF | Request permission

Abstract:

We investigate the solvability of a general class of mixed Hessian type equations on compact Kähler manifolds. Based on an observation in Guan and Zhang [Pure Appl. Math. Q. 17 (2021), pp. 865–907], the mixed Hessian equations can be rewritten as elliptic and concave equations which satisfy the general framework in Székelyhidi [J. Differential Geom. 109 (2018), pp. 337–378] and Phong and Tô [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), pp. 793–829] except one condition. We notice that the absence of this condition can be overcome by using the particular structure of the mixed Hessian equations. As an application, the solvability of the deformed Hermitian Yang-Mills equations in dimension three was also studied.

References

Zbigniew Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1735–1756 (English, with English and French summaries). MR 2172278, DOI 10.5802/aif.2137
Zbigniew Błocki, On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A 48 (2005), no. suppl., 244–247. MR 2156505, DOI 10.1007/BF02884710
Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359–372. MR 799272, DOI 10.1007/BF01389058
Xiuxiong Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 12 (2000), 607–623. MR 1772078, DOI 10.1155/S1073792800000337
Sławomir Dinew and Sławomir Kołodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math. 139 (2017), no. 2, 403–415. MR 3636634, DOI 10.1353/ajm.2017.0009
S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math. 3 (1999), no. 1, 1–15. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701920, DOI 10.4310/AJM.1999.v3.n1.a1
Hao Fang, Mijia Lai, and Xinan Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math. 653 (2011), 189–220. MR 2794631, DOI 10.1515/CRELLE.2011.027
Ji-Xiang Fu and Shing-Tung 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. MR 2396248
Ji-Xiang Fu and Shing-Tung Yau, A Monge-Ampère-type equation motivated by string theory, Comm. Anal. Geom. 15 (2007), no. 1, 29–75. MR 2301248, DOI 10.4310/CAG.2007.v15.n1.a2
Matt Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds, Comm. Anal. Geom. 19 (2011), no. 2, 277–303. MR 2835881, DOI 10.4310/CAG.2011.v19.n2.a2
Bo Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J. 163 (2014), no. 8, 1491–1524. MR 3284698, DOI 10.1215/00127094-2713591
Bo Guan and Wei Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 901–916. MR 3385185, DOI 10.1007/s00526-014-0810-1
Pengfei Guan and Xiangwen Zhang, A class of curvature type equations, Pure Appl. Math. Q. 17 (2021), no. 3, 865–907. MR 4278951, DOI 10.4310/PAMQ.2021.v17.n3.a2
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
Zuoliang Hou, Xi-Nan Ma, and Damin Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett. 17 (2010), no. 3, 547–561. MR 2653687, DOI 10.4310/MRL.2010.v17.n3.a12
N. V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc. 347 (1995), no. 3, 857–895. MR 1284912, DOI 10.1090/S0002-9947-1995-1284912-8
L. Chen, Hessian equations of Krylov type on Kähler manifolds, Preprint, arXiv:2107.12035, 2021.
Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, On estimates for the Fu–Yau generalization of a Strominger system, J. Reine Angew. Math. 751 (2019), 243–274. MR 3956695, DOI 10.1515/crelle-2016-0052
Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, The Fu-Yau equation with negative slope parameter, Invent. Math. 209 (2017), no. 2, 541–576. MR 3674222, DOI 10.1007/s00222-016-0715-z
Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, Fu-Yau Hessian equations, J. Differential Geom. 118 (2021), no. 1, 147–187. MR 4255073, DOI 10.4310/jdg/1620272943
Duong H. Phong and Dat T. Tô, Fully non-linear parabolic equations on compact Hermitian manifolds, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 3, 793–829 (English, with English and French summaries). MR 4311100, DOI 10.24033/asens.2471
Pingali, V., The deformed Hermitian Yang-Mills equation on three-folds, arXiv:1910.01870, 2019.
Jian Song and Ben Weinkove, On the convergence and singularities of the $J$-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61 (2008), no. 2, 210–229. MR 2368374, DOI 10.1002/cpa.20182
Wei Sun, Parabolic complex Monge-Ampère type equations on closed Hermitian manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3715–3733. MR 3426092, DOI 10.1007/s00526-015-0919-x
Wei Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^\infty$ estimate, Comm. Pure Appl. Math. 70 (2017), no. 1, 172–199. MR 3581825, DOI 10.1002/cpa.21652
Wei Sun, The parabolic flows for complex quotient equations, J. Geom. Anal. 29 (2019), no. 2, 1520–1545. MR 3935268, DOI 10.1007/s12220-018-0049-8
Gábor Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. 109 (2018), no. 2, 337–378. MR 3807322, DOI 10.4310/jdg/1527040875
Valentino Tosatti and Ben Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. MR 2669712, DOI 10.1090/S0894-0347-2010-00673-X
Ben Weinkove, Convergence of the $J$-flow on Kähler surfaces, Comm. Anal. Geom. 12 (2004), no. 4, 949–965. MR 2104082, DOI 10.4310/CAG.2004.v12.n4.a8
Ben Weinkove, On the $J$-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom. 73 (2006), no. 2, 351–358. MR 2226957
Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304
Jundong Zhou, A class of the non-degenerate complex quotient equations on compact Kähler manifolds, Commun. Pure Appl. Anal. 20 (2021), no. 6, 2361–2377. MR 4274388, DOI 10.3934/cpaa.2021085