Abstract
We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al. (J Reine Angew Math 616:89–130, 2008). This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity.
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Afuni, A. Local energy inequalities for mean curvature flow into evolving ambient spaces. manuscripta math. 158, 317–343 (2019). https://doi.org/10.1007/s00229-018-1027-y
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DOI: https://doi.org/10.1007/s00229-018-1027-y