Lagrangian mean curvature flow

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.

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.

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 $\rho =C\omega +nd{d}^{c}f$, 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.

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.

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\mathrm{Lag}}^{+}(n)$ of ${\mathbb{C}}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\mathrm{Lag}}^{+}(2)\simeq {\mathbb{S}}^1\times {\mathbb{S}}^2$. 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 $\Lambda$ on this region and prove that $\log \Lambda$ 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 $\Lambda$ must be a Lagrangian $n$-plane.

For each integer $K\ge 2$ when $n\ge 4$, and for $K=2,3,4$ when $n=3$, we construct an almost-calibrated Lagrangian mean curvature flow $L_K(t)$ in ${\mathbb{C}}^n$, 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 $$\underset{L_K(t)}{\sup }|{\mathbf{A}}_{L_K(t)}|\sim (T-t{)}^{-K/2}\text{as }t\nearrow T.$$ 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.

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 $C^2$-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated $A_{n-1}$-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, $\log \max |A(\cdot ,t)|$ is comparable to $\sqrt{t}$ as $t\to \infty$. 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.

Let $\mathcal{C}$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a\overline{L}$ which converge to $\mathcal{C}$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline{L}$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll 1$, 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.

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.

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.

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.

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|\Theta |\ge (n-2)\frac{\pi }{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^{\alpha }$ 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 $C^{2,\alpha }$ estimates by exponentiating the arctangent operator into a concave one when $|\Theta |\ge (n-2)\frac{\pi }{2}$ and $n>2$.

We study the self-shrinkers of the Legendrian mean curvature flow in the contact Euclidean space ${\mathbb{R}}^{2n+1}$, that is, the Legendrian submanifolds which move under the flow by the contact dilations $(x,y,z)\mapsto (\lambda x,\lambda y,{\lambda }^{2}z)$, 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 $|z|\le C$ is a Legendrian $n$-plane, and in particular there are no closed Legendrian self-shrinkers in ${\mathbb{R}}^{2n+1}$; this has no counterpart for Lagrangian self-shrinkers in ${\mathbb{C}}^n$, 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.

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 $n\ge 2$, 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.

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.

Let $L_t$ be a zero Maslov Lagrangian mean curvature flow in ${\mathbb{C}}^2.$ 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.

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.

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.

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.

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-Amp$\grave{e}$re type equation($\tau =0$) and the special Lagrangian parabolic equation($\tau =\frac{\pi }{2}$). Furthermore, we get the bound of $D^{l}u$, $l=\{3,4,5,\cdots \}$ for $\tau =\frac{\pi }{4}$ and the decay estimates of the higher order derivatives when $0<\tau <\frac{\pi }{4}$ and $\frac{\pi }{4}<\tau <\frac{\pi }{2}$. We also prove that $u(x,t)$ converges to smooth self-expanding solutions of.