Canonical foliation of bubblesheets

Abstract We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $$\mathbb R^k \times \mathbb {S}^{n-k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $$(n-k)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -spheres with QPMC.