\(C^{1,1}\) regularity of geodesics in the space of volume forms (Q2334516)

In [\textit{X. Chen} and \textit{W. He}, ``A class of fully nonlinear equations'', Preprint, \url{arXiv:1802.04985}], the authors considered a class of fully nonlinear equations which cover the equation introduced by \textit{S. K. Donaldson} a decade ago [in: The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press. 71--91 (2010; Zbl 1211.58015)] and the equation recently introduced by \textit{M. Gursky} and \textit{J. Streets} [``A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma_2\)-Yamabe problem'', Preprint, \url{arXiv:1603.07005}]. In the present paper, the author provides a \(\mathcal{C}^{1,1}\)-estimate for solutions for this class of equations and, as an application, it is proved the \(\mathcal{C}^{1,1}\)-regularity of geodesics in the space of volume forms.