Giovanni Bellettini, Carlo Mantegazza, Matteo Novaga
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.
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We develop estimates for the equation of scalar curvature of singular metrics with cone angle $β>1$, in a big and semi-positive cohomology class on a Kähler manifold. We further derive the Laplacian estimate for the scalar curvature equation of degenerate Kähler metrics. We then have several applications of these estimates on the singular constant scalar curvature Kähler metrics, which also include the singular Kähler-Einstein metrics.
We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.
In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N \ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.
We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension.
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small L^2-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the L^2-norm.
This paper is devoted to the construction of weak solutions to the singular constant $Q$-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of sufficiently large dimension with constant (positive) $Q$-curvature.
Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n = 2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than t…
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation.
In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.
In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our key ingredient is the existence result of general curvature flow with boundary conditions, which is proved in Section 4.
Decay order bound for mean curvature flow near compact singularities
针对形成一重数紧奇点的平均曲率流对应的重标度流，本文证明其衰减阶一致有界，进而得到相关唯一延拓结论。
Singular Ricci Flows on surfaces with boundary and positive scalar curvature
本文研究带边界紧致曲面上满足给定常值空间测地曲率的奇异 Ricci 流解的子序列收敛，并证明旋转对称情形下该收敛与边界测地曲率符号无关。
In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K ähler-Einstein surface. We show the relation between the maximum of the Kähler angle and the maximum of $|H|^2$ on the limit flow.
In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometr…
In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is solved when the singular Euler characteristic of the conical surface is non-positive.
Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow
本文研究带孤立锥形奇点的空间上的Ricci de Turck流，证明其保持Ricci曲率初始正则性与附加假设下的数量曲率正性，为奇异Ricci流的曲率算子正性研究奠定基础。
In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.
We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.
We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study the solution in a neighborhood of the blowup point, in some generic regimes, we find the key parameters take favorable signs and have sharp decay rates. We provide the remainder estimates in different norms.
In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of a dumbbell, which must develop a singularity in the flow before its area shrinks to zero. Simultaneously, by using the flow, we proved an isoperimetric inequality for some domains in $\mathbb{H}^3$.
Precise asymptotics near a generic $\mathbb S^1\times\mathbb R^3$ singularity of mean curvature flow
本文研究以气泡-薄片 $\mathbb S^1\times\mathbb R^3$ 为模型的一般平均曲率流奇点，并给出奇点邻域的精确渐近轮廓。
In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, using Perelman's $\mathcal{W}$-functional, we show that suitable blow-ups of our evolving metrics converge in the pointed Cheeger-Gromov sense to a Gaussian shrinker. If the flow has uniformly bounded scalar curvature and develops Type II singularities at $T$, we show that suitable…
Mean curvature flows with prescribed singular sets
研究构造了在R^{m+n}中相对于近欧几里得度量的光滑度量的均凸古老解，其首次奇点集恰好为K×{0}。
A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase
本文针对超临界相拉格朗日平均曲率流，证明奇点的内部先验估计，建立临界及超临界下成立的雅可比不等式，并推广至更广的拉格朗日平均曲率型方程。
We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isolated from the other ones.
A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature
本文通过局部研究曲率爆炸速率，发展了里奇流的精细奇点分析，证明了奇点只能是I型或II型，并应用于具有有界标量曲率的里奇流。
For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-Δ)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the exis…
Mean curvature flow through singularities
本文介绍平均曲率流基础，梳理三维空间中穿过奇点的平均曲率流精确理论的近年成果，阐述作者对四维空间非坍缩奇点的分类研究，并讨论相关开放问题与猜想。
We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.
Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lagrangian mean curvature flow.
We consider the problem of prescribing the $Q_{\ gamma}$ curvature on $\mathbb{S}^n$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant.
This paper is devoted to the Q-curvature type equation with singularities; mainly we give existence and regularity results of solutions. To have positive solutions which will be meaningfully in conformal geometry we restrict ourself to special manifolds.
In this paper, we give an example of a compact mean-convex hypersurface with a single singular point moved by mean curvature having a sequence of singular epochs (times) converging to zero.
Bol's type inequality for singular metrics and its application to prescribing $Q$-curvature problems
本文研究了奇异Liouville方程的高阶Bol不等式，并应用于规定Q曲率问题，推导了径向正常解存在的条件及总Q曲率的统一界。
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We p…
Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry
本文研究具有圆对称性的hyperkähler 4-流形中的拉格朗日平均曲率流，证明其可穿过有限时间奇点延续并收敛至special Lagrangians链，验证了Joyce猜想并指出奇点为颈缩。
We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $Σ$ admitting conical singularities of orders $α_i$'s at points $p_i$'s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $χ(Σ)+\sum_i α_i$ approaches a positive even integer, where $χ(Σ)$ is the Euler ch…
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an in…
This paper studies the self-similar singularity phenomenon of zero mean curvature equation including Born-Infeld equation, space-like surfaces with vanishing mean curvature equation and membrane equation which arises in string theory and geometric minimal surfaces theory. We show that there exist new explicit self-similar solutions for Born-Infeld equation, space-like surfaces with vanishing mean curvature equation and membrane equation in Minkowski space $R^{1+3}$ with respect to the radially symmetric case.
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