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<p>In his paper ‘Conjectures on Bridgeland Stability’ [<CitationRef CitationID="CR15">15</CitationRef>], Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore [<CitationRef CitationID="CR4">4</CitationRef>] answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of <i>manifolds with corners and a-corners</i> as introduced by Joyce [<CitationRef CitationID="CR16">16</CitationRef>]. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez [<CitationRef CitationID="CR21">21</CitationRef>].</p>

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Short-Time Existence of Lagrangian MCF Past Conical Singularities

  • 期刊论文

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  • 发表日期：

    2026年7月24日

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  • Spandan Ghosh

摘要

In his paper ‘Conjectures on Bridgeland Stability’ [15], Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore [4] answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce [16]. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez [21].