Abstract
We give sharp \(C^{2,\alpha }\) estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge–Ampère equations with conical singularities. As an application, we obtain a local estimate for Calabi–Yau equation in almost complex geometry. We also improve the \(C^{2,\alpha }\) regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.
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References
Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN 24, 5727–5766 (2013)
Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)
Calabi, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Mich. Math. J. 5, 105–126 (1958)
Chen, X.X., Wang, Y.Q.: On the long time behaviour of the conical Kähler Ricci flows (preprint). arXiv:1402.6689
Cherrier, P.: Équations de Monge–Ampère sur les variétés hermitiennes compactes. Bull. Sci. Math. (2) 111(4), 343–385 (1987)
Delanoë, P.: Sur l’analogue presque-complexe de l’équation de Calabi–Yau. Osaka J. Math. 33(4), 829–846 (1996)
Dinew, S., Kołodziej, S.: Liouville and Calabi–Yau type theorems for complex Hessian equations (preprint). arXiv:1203.3995
Dinew, S., Zhang, X., Zhang, X.: The \(C^{2,\alpha }\) estimate of complex Monge–Ampère equation. Indiana Univ. Math. J. 60(5), 1713–1722 (2011)
Fang, H., Lai, M., Ma, X.N.: On a class of fully nonlinear flows in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)
Guan, B., Li, Q.: Complex Monge–Ampère equations and totally real submanifolds. Adv. Math. 225(3), 1185–1223 (2010)
Guan, B., Sun, W.: On a class fully nonlinear elliptic equations on Hermitian manifolds (preprint). arXiv:1301.5863
Harvey, F.R., Lawson, H.B.: Potential theory on almost complex complex manifolds (preprint). arXiv:1107.2584
Li, Y.: A priori estimates for Donaldson’s equation over compact Hermitian manifolds. Calc. Var. Partial Differ. Equ. (to appear)
Lieberman, G.M.: Second order parabolic differential equations. World Scientific, London (1996)
Pliś, S.: The Monge–Ampère equation on almost complex manifolds (preprint). arXiv:1106.3356
Siu, Y.T.: DMV Seminar, vol. 8. Lectures on Herimitian–Einstein metrics for stable bundles and Kähler–Einstein metrics. Birkhäuser, Basel (1987)
Song, J., Weinkove, B.: On the convergence and singularities of the J-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61(2), 210–229 (2008)
Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds (preprint). arXiv:1310.0362
Tian, G.: On the existence of solutions of a class of Monge–Ampère equations. Acta Math. Sin. 4(3), 250–265 (1988)
Tian, G.: A third derivative estimate for conic Monge–Ampere equations (preprint)
Tosatti, V., Wang, Y., Weinkove, B., Yang, X.K.: \(C^{2,\alpha }\) estimate for nonlinear elliptic equations in complex and almost complex geometry Calc. Var. PDE (preprint, to appear). arXiv:1402.0554
Tosatti, V., Weinkove, B.: Estimates for the complex Monge–Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14(1), 19–40 (2010)
Tosatti, V., Weinkove, B.: The complex Monge–Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23(4), 1187–1195 (2010)
Tosatti, V., Weinkove, B.: The Monge–Ampère equation for \((n-1)\)-plurisubharmonic functions on a compact Kähler manifold (preprint). arXiv:1305.7511
Tosatti, V., Weinkove, B.: Hermitian metrics, \((n-1, n-1)\) forms and Monge–Ampère equations (preprint) arXiv:1310.6326
Tosatti, V., Weinkove, B., Yau, S.T.: Taming symplectic forms and the Calabi–Yau equation. Proc. Lond. Math. Soc. (3) 97(2), 401–424 (2008)
Trudinger, N.S.: Fully nonlinear, uniformly ellpic equations under natural structure conditions. Trans. Am. Math. Soc. 278(2), 751–769 (1983)
Wang, L.H.: On the regularity theory of fully nonlinear parabolic equations: II. Commun. Pure Appl. Math. 45(2), 141–178 (1992)
Wang, Y.: On the \(C^{2,\alpha }\)-regularity of the complex Monge–Ampère equation. Math. Res. Lett. 19(4), 939–946 (2012)
Weinkove, B.: Convergence of the J-flow on Kähler surfaces. Commun. Anal. Geom. 12(4), 949–965 (2004)
Weinkove, B.: On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73(2), 351–358 (2006)
Weinkove, B.: The Calabi–Yau equation on almost-Kähler four-manifolds. J. Differ. Geom. 76(2), 317–349 (2007)
Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Apmère equation I. Commun. Pure Appl. Math. 31, 339–411 (1978)
Acknowledgments
The author would like to thank his advisor Gang Tian for constant encouragement and several useful comments on an earlier version of this paper. The author would also like to thank Yuanqi Wang for some helpful discussions about complex Monge–Ampère equations with conical singularities. The author would also like to thank Valentino Tosatti and Jingang Xiong for many helpful conversations. The author would also like to thank CSC for supporting the author visiting Northwestern University.
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Communicated by A. Chang.
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Chu, J. \(C^{2,\alpha }\) regularities and estimates for nonlinear elliptic and parabolic equations in geometry. Calc. Var. 55, 8 (2016). https://doi.org/10.1007/s00526-015-0948-5
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DOI: https://doi.org/10.1007/s00526-015-0948-5