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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb R^k \times \mathbb {S}^{n-k}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="double-struck">R</mi> <mi>k</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n-k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spheres with QPMC.</p>

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Canonical foliation of bubblesheets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月4日

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  • Jean Lagacé,
  • Stephen Lynch

摘要

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 \(\mathbb R^k \times \mathbb {S}^{n-k}\) R k × S n - k , equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded \((n-k)\) ( n - k ) -spheres with QPMC.