Andrew A. Cooper
In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds are characterised by the blow up of the mean curvature.
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The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow
本文证明拉格朗日平均曲率流在非退化颈缩奇点处平均曲率无界，并给出爆破速率控制。
We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our result generalizes an earlier result of the author and of Hoffman and White.
In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.
We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the x_3-axis.
In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given.
For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.
Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination of U\K. Assuming results of Colding-Minicozzi and an extension due to Meeks, we prove that such a blow-up set K must be contained in a C^1 embedded curve.
Under certain conditions such as the $2$-convexity, a singularity of the level set flow is of type I (in the sense that the rate of curvature blow-up is constrained before and after the singular time) if and only if the flow shrinks to either a round point or a $C^{1}$ curve near that singular point. Analytically speaking, the arrival time is $C^{2}$ near a critical point if and only if it satisfies a Lojasiewicz inequality near the point.
For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.
Blow-up sets of Ricci curvatures of complete conformal metrics
本文研究完备共形度量在边界接近低维极限集时 Ricci 曲率的爆破现象，依据 Yamabe 不变量刻画爆破集，并证明所得结果的最优性。
Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation (after Chen and Wu)
针对常标量曲率与常边界平均曲率方程，本文将低能量解集失去紧性的光滑反例维数阈值由 Chen 和 Wu 的不小于62维改进至不小于35维。
Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$", with respect to initial data. We employ an Angenent-like neck-pinching argument to force singularities in nearby flows. However, since we cannot prescribe initial data, we combine Andrews' $α$-non-collapsed condition and Colding and Minicozzi's uniqueness of tangent flows to place appropriately sized spheres in the region inside the hypersurface.
The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces
In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in $\bbr^n$, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds i…
Blowing up sequences of constant mean curvature tori in $\mathbb{R}^3$ to minimal surfaces
本文研究通过爆破有限型常平均曲率平面序列得到极小曲面浸入的条件，并将其转化为从 sinh-Gordon 可积系统解爆破得到 Liouville 方程解，揭示后者与 KdV 可积系统的联系。
In this short paper, we show that Kähler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption.
Scalar curvature of blow-ups of compact Kähler manifolds along complex submanifolds
针对复维数至少3且含复余维数至少2子流形的紧凯勒流形，证明其沿该子流形爆破所得流形上存在凯勒度量列，其纯量曲率按$C^0$范数收敛到原度量拉回的纯量曲率，推广了点爆破的对应结果。
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.
In this paper we study the blow up sequence of mean curvature flow of surfaces in $\mathbb R^3$ with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with finite entropy is a smoothly embedded self-shrinker.
We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate $(T-t)^{-1}$. (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infi…
We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails when the dimension of the manifold is not less than $62$.
Nonnegative curvature on vector bundles over ${CP}^3$ and its blow-up
本文证明 $CP^3$ 上秩至少七、$CP^3\#CP^3$ 上秩至少八的实向量丛及其单位球面丛具有非负截面曲率的完备度量，通过低维丛分类与 Grove--Ziller 提升理论实现。
Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the same maximum and barrier principles that a smooth minimal submanifold satisfies. For suitable open subsets W of N, this allows one to show that if the areas of the M_i are uniformly bounded on compact subsets of W, then the areas are in fact uniformly bounded on all compact subset…
Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.
The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $Σ$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\timesΣ$ and a family of metrics $γ(r)$ on $Σ$, when the coefficients of this equation are appropriately defined in terms of $γ$ and $R$, positive solutions give metrics of prescribed scalar curvature $R$ on $[r_0,r_1)\timesΣ$ in the form \[ g=u^2dr^2+r^2γ.\] If the area element of $r^2γ$ is expanding for increasing $r$, then the equation is parabolic, and the basic existence problem is to take p…
We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.
In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.
Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curvatures and injectivity radii. In particular we prove blow up criteria for the harmonic Ricci flow and for the spinor flow on surfaces.
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.
In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow.…
In this paper, we shall prove that, on a non-flat Riemannian vector bundle over a compact Riemannian manifold, the smooth solution of the Yang-Mills flow will blow up in finite time if the energy of the initial connection is small enough. We also consider the finite time blow up for the Yang-Mills flow with the initial curvature near the harmonic form. Furthermore, when $E$ is a holomorphic vector bundle over a compact Kähler manifold, then $E$ will admit a projective flat structure if the trace free part of Chern curvature is small enough.
Uniqueness of blowups for forced mean curvature flow
本文证明了受迫平均曲率流在闭自收缩子和圆圆柱自收缩子处切锥的唯一性，推广了Schulze和Colding-Minicozzi的结果，并涵盖黎曼流形情形。
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $γ>1/2$, the…
Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature
本文通过构造新泛函研究幂零主丛上里奇流 blowdown 极限的刚性，证明三维海森堡群情形下极限为扩张里奇孤子，并结合 Lott 的阿贝尔结果完成四维情形的完整局部分类。
Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature
本文沿用Velázquez、Guo、Sesum和Stolarski的论证方法，在$\mathbb{R}^n$（$n\geq 8$）中构造了具平均曲率有界的紧致平均曲率流解。
We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is known that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci flow along the exceptional divisor is the unique $U(n)$-complete shrinking Kähler-Ricci soliton on $\mathbb C^n$ blown-up at one point. This establishes the conjecture of Feldman-Ilmanen-Knopf.
In this paper, we consider ancient noncollapsed mean curvature flows $M_t=\partial K_t\subset \mathbb{R}^{n+1}$ that do not split off a line. It follows from general theory that the blowdown of any time-slice, $\lim_{λ\to 0} λK_{t_0}$, is at most $n-1$ dimensional. Here, we show that the blowdown is in fact at most $n-2$ dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly $k$-convex case, the blowdown is at most $k-2$ dimensional. This generalizes recent resu…
In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of (Invent. Math., 1987).
In this short note, we study the behavior of Kaher-Ricci flow on Kahler manifolds which contract divisors to smooth submanifolds. We show that the Kahler potentials are Holder continuous and the flow converges sequentially in Gromov-Hausdorff topology to a compact metric space which is homeomorphic to the base manifold.
We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow $ω_t$, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time $T<\infty $. We prove that the scalar curvature of $ω_t$ is bounded from above by $C/(T-t)^2$ under the existence of a contraction associated to the limiting cohomology class $[ω_T]$. This generalizes Z. Zhang's work to the conic case.
Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions
本文证明在维数≥25且具有正Yamabe不变量的闭流形上，存在C^1-接近的度量，其共形类中有无穷多个具有相同常Q-曲率和大能量的光滑度量，以及一个具有无界体积的常Q-曲率度量序列。
No compact split limit Ricci flow of type II from the blow-down
本文利用Perelman的$\mathcal L$-测地线理论，证明了在满足特定曲率条件的非紧$κ$-非坍缩稳态梯度Ricci孤立子中，由blow-down产生的余维一紧分裂古老解必为I型，并将此结论推广至任意维度。
Scalar curvature on Kähler blow-ups and systolic inequalities
本文发展针对特定Kähler流形的加权水平集方法，证明其爆破的Kähler度量序列的纯量曲率逼近原度量，建立正纯量曲率Kähler流形的精确2-收缩估计及偶收缩不等式。
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 show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example $\int_{0}^{t} \int_{M_{s}} \frac{\abs{A}^{n + 2}}{log (2 + \abs{A})} dμds,$ blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.
We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.
We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a la…
We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when the regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.
We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class …
The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.
Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup…
We study the black hole interiors of spacetimes arising from gravitational collapse in the spherically symmetric Einstein-scalar field setting, and we investigate the precise blow-up rates of curvature and mass at the spacelike singularity near timelike infinity. We show in particular that the Kretschmann scalar blows up faster than in the Schwarzschild setting, due to mass inflation. Moreover, the blow-up rate is not constant and converges to the Schwarzschild rate towards timelike infinity and it depends on the precise late-time polynomial behaviour of the scalar field along the event horizo…
Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we classify Type II blow-ups of almost calibrated Lagrangian mean curvature flow when the blow-down is a pair of transverse planes or, when n=2, a multiplicity two plane. We also show that the Harvey-Lawson Clifford torus cone in C^3 cannot arise as the blow-down of an almost calibra…
A blow-up method to prescribed mean curvature graphs with fixed boundaries
本文运用Schoen和Yau的爆破方法，结合曲率估计与Nc-f条件，解决了黎曼流形上预定平均曲率Dirichlet问题，并应用于PMC Plateau问题。
Geometric regularity of blow-up limits of the Kähler-Ricci flow
本文证明了 Kähler-Ricci 流 Type I 爆破极限的几何正则性，得到极限流在 Gromov-Hausdorff 和 Gromov-$W_1$ 距离下时间连续，且奇异集余维数至少为 4。
We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]).
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.
We establish a pointwise estimate of A along the mean curvature flow in terms of the initial geometry and the jHAj bound. As corollaries we obtain the extension theorem of HA and the blowup rate estimate of HA.
In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.
We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting planes, any Type II blowup must be the Lawlor neck with the same asymptotes, and these blowups are independent of the choice of rescaling. We also give a partial classification of when singularities occur in the equivariant case, and examine the intermediate scales between the Typ…
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