Contrast to the hyperbolic mean curvature flows studied in "Hyperbolic mean curvature flow," J. Differential Equations 246 (2009), 373–390, "The hyperbolic mean curvature flow," J. Math. Pures Appl. 90 (2008) 591–614, and "Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow," J. Differential Equations 247 (2009) 1694–1719, a new hyperbolic curvature flow is proposed for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. The equation satisfied by the graph of the hypersurface under this flow gives rise to a new class of fully nonlinear Euclidean invariant hyperbolic equations.
"On hyperbolic Gauss curvature flows." J. Differential Geom. 89 (3) 455 - 485, November 2011. https://doi.org/10.4310/jdg/1335207375