Abstract
We prove a \(C^{1,1}\) estimate for solutions of a class of fully nonlinear equations introduced by Chen–He. As an application, we prove the \(C^{1,1}\) regularity of geodesics in the space of volume forms.
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Błocki, Z.: Regularity of the degenerate Monge–Ampère equation on compact Kähler manifolds. Math. Z. 244, 153–161 (2003)
Chen, X.X., He, W.Y.: The space of volume forms. Int. Math. Res. Not. IMRN 5, 967–1009 (2011)
Chen, X.X., He, W.Y.: A class of fully nonlinear equations. Preprint arXiv:1802.04985
Chu, J., Tosatti, V., Weinkove, B.: On the \(C^{1,1}\) regularity of geodesics in the space of Kähler metrics. Ann. PDE 3(2), Art. 15, (2017)
Darvas, T.: Morse theory and geodesics in the space of Kähler metrics. Proc. Am. Math. Soc. 142(8), 2775–2782 (2014)
Darvas, T., Lempert, L.: Weak geodesics in the space of Kähler metrics. Math. Res. Lett. 19(5), 1127–1135 (2012)
Donaldson, S.: Nahm’s Equations and Free-Boundary Problems, The Many Facets of Geometry, pp. 71–91. Oxford University Press, Oxford (2010)
Gursky, M., Streets, J.: A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma _{2}\)-Yamabe problem. Geom. Topol. 22(6), 3501–3573 (2018)
He, W.Y.: The Donaldson equation. Preprint arXiv:0810.4123
Lempert, L., Vivas, L.: Geodesics in the space of Kähler metrics. Duke Math. J. 162(7), 1369–1381 (2013)
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Communicated by A.Chang.
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Chu, J. \(C^{1,1}\) regularity of geodesics in the space of volume forms. Calc. Var. 58, 194 (2019). https://doi.org/10.1007/s00526-019-1648-3
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DOI: https://doi.org/10.1007/s00526-019-1648-3