<p>In this paper, we consider ancient noncollapsed mean curvature flows <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M_t=\partial K_t\subset \mathbb {R}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>∂</mi> <msub> <mi>K</mi> <mi>t</mi> </msub> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> that do not split off a line. It follows from general theory that the blowdown of any time slice, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lim _{\lambda \rightarrow 0} \lambda K_{t_0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </msub> <mi>λ</mi> <msub> <mi>K</mi> <msub> <mi>t</mi> <mn>0</mn> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation>, is at most <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> dimensional. Here, we show that the blowdown is in fact at most <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly <i>k</i>-convex case, the blowdown is at most <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> dimensional. This generalizes the recent results from Choi et al. (Geom. Topol. 28(7):3095–3134, 2024). The results and methods developed in this paper and its sequel by Du and Zhu (Adv. Math. 479:110422, 2025) have several applications in the study of mean curvature flow through singularities in higher dimensions.</p>

The Blowdown of Ancient Noncollapsed Mean Curvature Flows

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发表日期：

2026年2月12日

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Wenkui Du,
Robert Haslhofer

摘要

In this paper, we consider ancient noncollapsed mean curvature flows $M_t=\partial K_t\subset \mathbb {R}^{n+1}$ $M_{t} = \partial K_{t} \subset R^{n + 1}$ that do not split off a line. It follows from general theory that the blowdown of any time slice, $\lim _{\lambda \rightarrow 0} \lambda K_{t_0}$ ${lim}_{λ \to 0} λ K_{t_{0}}$ , is at most $n-1$ $n - 1$ dimensional. Here, we show that the blowdown is in fact at most $n-2$ $n - 2$ dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly k-convex case, the blowdown is at most $k-2$ $k - 2$ dimensional. This generalizes the recent results from Choi et al. (Geom. Topol. 28(7):3095–3134, 2024). The results and methods developed in this paper and its sequel by Du and Zhu (Adv. Math. 479:110422, 2025) have several applications in the study of mean curvature flow through singularities in higher dimensions.