Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$

We consider the evolution by mean curvature flow of a closed sub-manifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. The latter behavior is only possible if the dimension is even. These results generalize previous works by Huisken and Baker on the mean curvature flow of submanifolds of the sphere.

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The results of this paper are part of Giuseppe Pipoli’s PhD thesis, written at the Department of Mathematics, University “Sapienza” of Rome. Giuseppe Pipoli was partially supported by PRIN07 “Geometria Riemanniana e strutture differenziabili” of MIUR (Italy) and Progetto universitario Univ. La Sapienza “Geometria differenziale – Applicazioni”. Carlo Sinestrari was partially supported by FIRB–IDEAS project “Analysis and beyond” and by the group GNAMPA of INdAM (Istituto Nazionale di Alta Matematica).

Received 3 February 2015

Published 1 November 2017