https://neckpinch.com/topics/lagrangian-mean-curvature-flow/New arXiv papers on lagrangian mean curvature flow, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.34923arxiv:2609.34923Mon, 28 Sep 2026 00:00:00 GMTCarlos Alberto Ochoa Flores — A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauLong-time existence & convergence https://arxiv.org/abs/2609.31343arxiv:2609.31343Fri, 25 Sep 2026 00:00:00 GMTJason D. Lotay, Goncalo Oliveira — We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2609.23627arxiv:2609.23627Sun, 20 Sep 2026 00:00:00 GMTNaotoshi Fujihara, Toru Kajigaya, Albert Wood — We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies ρ= Cω+ ndd^cf, and which satisfy C>0. With analogous assumptions on the flow, we obtain subconvergence to an f-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that (M,g,f) is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.Mean curvature flowCurve shorteningLagrangian mean curvature flowEinstein metricsSingularitiesSurfacesKähler geometryHomogeneous spacesLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2609.19724arxiv:2609.19724Thu, 17 Sep 2026 00:00:00 GMTPing-Hung Lee — We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.Ricci flowMean curvature flowLagrangian mean curvature flowSurfacesScalar curvatureLong-time existence & convergenceDiscrete & graph Ricci https://arxiv.org/abs/2608.14444arxiv:2608.14444Fri, 14 Aug 2026 00:00:00 GMTTang-Kai Lee, Mu-Tao Wang — We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian rm Lag⁺(n) of ℂⁿ, an intrinsic higher-dimensional analog of a natural convex region in rm Lag⁺(2) ≃ S¹× S². A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function Λ on this region and prove that log Λ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for Λ must be a Lagrangian n-plane.Mean curvature flowLagrangian mean curvature flowStability & uniqueness https://arxiv.org/abs/2607.03152arxiv:2607.03152Fri, 03 Jul 2026 00:00:00 GMTMaxwell Stolarski, Wei-Bo Su — For each integer K≥2 when n≥4, and for K=2,3,4 when n=3, we construct an almost-calibrated Lagrangian mean curvature flow L_K(t) in ℂⁿ, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time T with the explicit curvature blow up rate sup_L_K(t) |A_L_K(t)| ∼ (T-t)^(−K/2) as t↗ T . The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.Mean curvature flowLagrangian mean curvature flowSingularitiesHomogeneous spacesEigenvalues & spectrum https://arxiv.org/abs/2606.28767arxiv:2606.28767Sat, 27 Jun 2026 00:00:00 GMTPing-Hung Lee, Chung-Jun Tsai — We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons–Hawking spaces. We consider circle-invariant Lagrangian 2-spheres whose quotient curves are concave and are C²-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated Aₙ₋₁-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, logmax |A( · ,t)| is comparable to √t as t→∞. The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2606.21541arxiv:2606.21541Fri, 19 Jun 2026 00:00:00 GMTMaxwell Stolarski, Wei-Bo Su — Let 𝓒 be a G-invariant special Lagrangian cone admitting a scaled family of G-invariant special Lagrangian desingularizations a overline L which converge to 𝓒 as a↘ 0. We study the linearized self-shrinker operator on aoverline L in a Gaussian weighted L² space of G-equivariant functions. For 0<a≪1, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesEigenvalues & spectrum https://arxiv.org/abs/2603.20498arxiv:2603.20498Fri, 20 Mar 2026 00:00:00 GMTArunima Bhattacharya, Micah Warren, Daniel Weser — We express the mean curvature flow of Lagrangian submanifolds in pseudo-Riemannian manifolds endowed with the Kim-McCann-Warren metric within the framework of generalized mean curvature flow on Kim-McCann manifolds. While generalized mean curvature flow has been studied in Kähler geometry, our work shows that techniques from para-Kähler geometry arise naturally in the Kim-McCann setting. Using this perspective, we prove that the Lagrangian condition is preserved along the flow. By identifying generalized mean curvature flow with Lagrangian mean curvature flow, we show that the Ma-Trudinger-Wang regularity theory applies to this setting. In particular, the cross-curvature positivity condition of Kim-McCann yields smoothly converging flows of Lagrangian submanifolds. Under the cross-curvature condition, any Lagrangian submanifold avoiding the cut locus converges exponentially to a stationary submanifold, which locally arises as the graph of an optimal transport map. Our framework substantiates the analogy between special Lagrangian geometry in almost Calabi-Yau manifolds and optimal transport theory in the Kim-McCann setting. In particular, we show that Kim-McCann manifolds equipped with a para-holomorphic volume form serve as the natural counterpart to almost Calabi-Yau manifolds.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauKähler geometryLimit spaces & RCDLong-time existence & convergence https://arxiv.org/abs/2603.02749arxiv:2603.02749Tue, 03 Mar 2026 00:00:00 GMTJacopo Stoppa — As part of his work on special Lagrangian (sLag) submanifolds with isolated conical singularities, Joyce proved a criterion for the existence of sLag smoothings, along a small variation of complex structure, for the union of two connected, compact, embedded sLags, with the same phase, intersecting transversely. Here we construct infinitely many examples of pairs of non-compact, embedded sLags, of the same phase and with arbitrary dimension, intersecting only at infinity in a non-transverse way, which satisfy Joyce's criterion: along a small variation of complex structure, a sLag smoothing of their union exists on the stable locus where a slope inequality for periods of the holomorphic volume form holds. At least under a natural symmetry assumption, this slope inequality is also necessary for the existence of such smoothing. Our approach uses the Leung-Yau-Zaslow transform and the analysis of deformed Hermitian Yang-Mills connections with Calabi ansatz, due to Jacob and Sheu. In the unstable case, we prove that if a family of Lagrangian smoothings evolving under the natural Calabi-symmetric version of the mean curvature flow (due to Chan and Jacob) admits a limit, then this must be the union of the original sLags. As an application we show that in our examples, in dimension two, the condition for the existence of the sLag smoothing is in fact equivalent to the stability of the corresponding object in the Fukaya-Seidel category, with respect to a known Bridgeland stability condition imported from algebraic geometry, and in the unstable case the limit of the Calabi-symmetric mean curvature flow in our result coincides with the Harder-Narasimhan decomposition, consistently with a general conjecture of Joyce. A similar (although weaker) result also holds in dimension three.Mean curvature flowLagrangian mean curvature flowSingularities3-manifoldsKähler geometryStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2602.15771arxiv:2602.15771Tue, 17 Feb 2026 00:00:00 GMTGábor Székelyhidi — We introduce a notion of nondegenerate neck pinch singularity along the Lagrangian mean curvature flow of surfaces in a Calabi-Yau surface. We show that such singularities can occur, are stable under small perturbations, and any neck pinch singularity can be perturbed to such a nondegenerate singularity near the singular time. Using this we answer some questions raised by Neves and Joyce. We also introduce nondegenerate teardrop singularities and show that these cannot occur for embedded flows.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSingularitiesKähler geometry https://arxiv.org/abs/2510.22741arxiv:2510.22741Sun, 26 Oct 2025 00:00:00 GMTArunima Bhattacharya, Ravi Shankar, Jeremy Wall, Diego Yepez — We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., |Θ|≥ (n-2)tfracπ2, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct C^α singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving C^(2,α) estimates by exponentiating the arctangent operator into a concave one when |Θ|≥ (n-2)tfracπ2 and n>2.Mean curvature flowLagrangian mean curvature flowSingularitiesWeak & singular flows https://arxiv.org/abs/2508.07900arxiv:2508.07900Mon, 11 Aug 2025 00:00:00 GMTShu-Cheng Chang, Hongbing Qiu, Liuyang Zhang — We study the self-shrinkers of the Legendrian mean curvature flow in the contact Euclidean space ℝ²ⁿ⁺¹, that is, the Legendrian submanifolds which move under the flow by the contact dilations (x,y,z)↦(λx,λy,λ²z), and we explain why it is the generator of these dilations, and not the Euclidean position vector, which appears in the self-shrinker equation. Our main result is that a complete properly immersed Legendrian self-shrinker with bounded Legendrian angle is a Legendrian n-plane. On a Legendrian self-shrinker the Legendrian angle is a constant multiple of the coordinate z, so that this is a statement about the position of the submanifold, and it has two consequences. A complete properly immersed Legendrian self-shrinker which lies in a slab |z|≤ C is a Legendrian n-plane, and in particular there are no closed Legendrian self-shrinkers in ℝ²ⁿ⁺¹; this has no counterpart for Lagrangian self-shrinkers in ℂⁿ, among which there are closed ones. And the potential of an entire smooth Legendrian self-shrinking graph is a quadratic polynomial, without any further assumption. The proof uses a volume growth estimate for Legendrian self-shrinkers with respect to a Gaussian weight which is built from the horizontal part of the generator of the dilations.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2507.16129arxiv:2507.16129Tue, 22 Jul 2025 00:00:00 GMTJiguang Bao, Zixiao Liu — This paper investigates the asymptotic behavior at infinity of ancient solutions to the Lagrangian mean curvature flow. Under conditions that admit Liouville type rigidity theorems, we prove that every classical solution converges at infinity to the sum of a quadratic polynomial in x and a linear function in t, with an explicitly derived exponential rate of convergence. As a critical part of the proof framework of this paper, we establish the existence of a global viscosity solution with prescribed asymptotic behavior at infinity, featuring two key innovations: (i) applicability to all dimensions n≥ 2, and (ii) no requirement that the Hessian matrix of the prescribed quadratic term be positive definite or close to a scalar multiple of the identity matrix. These results establish the relationship between Liouville type rigidity, asymptotic analysis at infinity, and the existence of viscosity solutions.Mean curvature flowLagrangian mean curvature flowAncient solutionsWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2507.10341arxiv:2507.10341Mon, 14 Jul 2025 00:00:00 GMTSpandan Ghosh — In his paper `Conjectures on Bridgeland Stability', 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 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. 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.Mean curvature flowLagrangian mean curvature flowRicci solitonsWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trickPhase fields & multiphase flows https://arxiv.org/abs/2503.22918arxiv:2503.22918Sat, 29 Mar 2025 00:00:00 GMTSourav Ghosh — Let Lₜ be a zero Maslov Lagrangian mean curvature flow in ℂ². We show that if the mean curvature stays uniformly bounded along the flow, then the tangent flow at a singular point is unique i.e. the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2412.15880arxiv:2412.15880Fri, 20 Dec 2024 00:00:00 GMTXiang Li, Yong Luo, Jun Sun — In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesSurfacesStability & uniqueness https://arxiv.org/abs/2410.22172arxiv:2410.22172Tue, 29 Oct 2024 00:00:00 GMTYang Li, Gábor Székelyhidi — We study singularities along the Lagrangian mean curvature flow with tangent flows given by multiplicity one special Lagrangian cones that are smooth away from the origin. Some results are: uniqueness of all such tangent flows in dimension two; uniqueness in any dimension when the link of the cone is connected; the existence of nontrivial special Lagrangian blowup limits. We also prove a singular version of Imagi-Joyce-dos Santos's uniqueness result of the Lawlor neck. As an application we prove that in any dimension, singularities that admit a tangent flow given by the union of two transverse planes is modeled on shrinking Lawlor necks at suitable scales.Mean curvature flowLagrangian mean curvature flowSingularitiesStability & uniqueness https://arxiv.org/abs/2410.17850arxiv:2410.17850Wed, 23 Oct 2024 00:00:00 GMTXiaoli Han, Jiayu Li, Jun Sun — It is known that there is no a Type I singularity for the Lagrangian mean curvature flow with zero Maslov class. In this paper, we study translating solitons which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we try to understand the important open question proposed by Joyce-Lee-Tsui and Neves-Tian, whether the Lagrangian translating solitons constructed by Joyce-Lee-Tsui can be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularities https://arxiv.org/abs/2410.17794arxiv:2410.17794Wed, 23 Oct 2024 00:00:00 GMTShanshan Li, Jiaru Lv, Rongli Huang — In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation has a smooth solution u(x,t) for three corresponding nonlinear equations between the Monge-Ampgraveere type equation(τ=0) and the special Lagrangian parabolic equation(τ=fracπ2). Furthermore, we get the bound of D^lu, l=\3,4,5,⋯\ for τ=fracπ4 and the decay estimates of the higher order derivatives when 0<τ<fracπ4 and fracπ4<τ<fracπ2. We also prove that u(x,t) converges to smooth self-expanding solutions of.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2407.12756arxiv:2407.12756Wed, 17 Jul 2024 00:00:00 GMTArunima Bhattacharya, Jeremy Wall — In this paper, we prove interior a priori estimates for singularities of the Lagrangian mean curvature flow assuming the Lagrangian phase is supercritical. We prove a Jacobi inequality that holds good when the Lagrangian phase is critical and supercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2407.12733arxiv:2407.12733Wed, 17 Jul 2024 00:00:00 GMTArunima Bhattacharya, Micah Warren, Daniel Weser — We prove a Liouville type result for convex solutions of the Lagrangian mean curvature flow with restricted quadratic growth assumptions at antiquity on the solutions.Mean curvature flowLagrangian mean curvature flowAncient solutions https://arxiv.org/abs/2406.04602arxiv:2406.04602Fri, 07 Jun 2024 00:00:00 GMTPing-Hung Lee, Chung-Jun Tsai — It is known that minimal Lagrangians in Kähler–Einstein manifolds of non-positive scalar curvature are linearly stable under Hamiltonian deformations. We prove that they are also stable under the Lagrangian mean curvature flow, and therefore establish the equivalence between linear stability and dynamical stability. Specifically, if one starts the mean curvature flow with a Lagrangian which is C¹-close and Hamiltonian isotopic to a minimal Lagrangian, the flow exists smoothly for all time, and converges to that minimal Lagrangian. Due to the work of Neves [Ann. of Math. 2013], this cannot be true for C⁰-closeness.Mean curvature flowLagrangian mean curvature flowEinstein metricsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2406.04591arxiv:2406.04591Fri, 07 Jun 2024 00:00:00 GMTXishen Jin, Jiawei Liu — In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in Smoczyk-Tsui-Wang's work. More precisely, we prove that if the graph induced by a closed 1-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.Mean curvature flowLagrangian mean curvature flowStability & uniqueness https://arxiv.org/abs/2405.11049arxiv:2405.11049Fri, 17 May 2024 00:00:00 GMTTristan C. Collins, Adam Jacob, Yu-Shen Lin — We establish a convergence result for the mean curvature flow starting from a totally real submanifold which is "almost minimal" in a precise, quantitative sense. This extends, and makes effective, a result of H. Li for the Lagrangian mean curvature flow.Mean curvature flowLagrangian mean curvature flow https://arxiv.org/abs/2403.07235arxiv:2403.07235Tue, 12 Mar 2024 00:00:00 GMTArunima Bhattacharya, Jeremy Wall — In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.Mean curvature flowLagrangian mean curvature flowRicci solitonsSelf-shrinkers & translators https://arxiv.org/abs/2401.02228arxiv:2401.02228Thu, 04 Jan 2024 00:00:00 GMTWei-Bo Su, Chung-Jun Tsai, Albert Wood — In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as t→∞. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas–Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family N^(ε(t)) constructed by gluing in special Lagrangian `Lawlor necks' of size ε(t), where the dynamics of the neck size ε(t) are driven by the obstruction for the existence of nearby special Lagrangians to N^(ε(t)). This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.Ricci flowMean curvature flowLagrangian mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2312.15996arxiv:2312.15996Tue, 26 Dec 2023 00:00:00 GMTShu-Cheng Chang, Chin-Tung Wu, Liuyang Zhang — In this article, we recapture the Smale conjecture on a Sasakian 3-sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian 3-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in S²× S³. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in S²× S³. Finally, we will address the rigidity theorem of 2-dimensional Legendrian self-shrinkers in ℝ⁵. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ℂ³ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesSmale conjecture & diffeomorphismsStability & uniquenessSasakian & contact geometry https://arxiv.org/abs/2310.17709arxiv:2310.17709Thu, 26 Oct 2023 00:00:00 GMTYu Hin Chan, Adam Jacob — The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this paper we construct two distinct examples of singularities along the flow. First, we find a finite time singularity, ruling out long time existence of the flow in general. Next we show long time existence of the flow with a Calabi symmetry assumption on the blowup of ℙⁿ, n≥ 3, if one assumes supercritical phase. Using this, we find an example where a singularity occurs at infinite time along the destabilizing subvariety in the semi-stable case.Mean curvature flowLagrangian mean curvature flowSingularitiesKähler geometryLong-time existence & convergence https://arxiv.org/abs/2309.09432arxiv:2309.09432Mon, 18 Sep 2023 00:00:00 GMTChung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang — Given an entire C² function u on ℝⁿ, we consider the graph of D u as a Lagrangian submanifold of ℝ²ⁿ, and deform it by the mean curvature flow in ℝ²ⁿ. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of (I+(D² u)²)⁻¹D² u. Such results were previously known only under the stronger assumption of positivity of D² u.Mean curvature flowLagrangian mean curvature flowLong-time existence & convergence