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May 26, 2017 at 19:22 comment added Thisquestionisreallyhard Thanks! I realize that in the Kahler setting I only proved it if the vector field is further assumed gradient (which I think is still correct). If I've understood your example, it possibly suggests a more general statement: namely that in the setting I describe there exists a one-parameter family $\psi_t$ of isometries of $g_t = \phi_t^* g_0$ such that $\phi_t \circ \psi_t$ are biholomorphisms?
May 26, 2017 at 19:08 history answered Robert Bryant CC BY-SA 3.0