Suppose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(M^i_t)_{t\in [0,T)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mi>t</mi> <mi>i</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, are two mean curvature flows in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> encountering a multiplicity one compact singularity at time T, in such a manner that for every k, the Hausdorff distance between the two flows, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation>, satisfies <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d_{H}(M^1_t,M^2_t)/(T-t)^k \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>H</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mi>t</mi> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>M</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We demonstrate that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M^1_t=M^2_t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>M</mi> <mi>t</mi> <mn>1</mn> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for every t. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M^1_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mi>t</mi> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> is itself a self-similarly shrinking flow.

How close is too close for singular mean curvature flows?

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2026年7月18日

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J. M. Daniels-Holgate,
Or Hershkovits

摘要

Suppose $(M^i_t)_{t\in [0,T)}$ ${(M_{t}^{i})}_{t \in [0, T)}$ , $i=1,2$ $i = 1, 2$ , are two mean curvature flows in $\mathbb {R}^{n+1}$ $R^{n + 1}$ encountering a multiplicity one compact singularity at time T, in such a manner that for every k, the Hausdorff distance between the two flows, $d_H$ $d_{H}$ , satisfies $d_{H}(M^1_t,M^2_t)/(T-t)^k \rightarrow 0$ $d_{H} (M_{t}^{1}, M_{t}^{2}) / {(T - t)}^{k} \to 0$ . We demonstrate that $M^1_t=M^2_t$ $M_{t}^{1} = M_{t}^{2}$ for every t. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where $M^1_t$ $M_{t}^{1}$ is itself a self-similarly shrinking flow.