Julie Clutterbuck, Oliver C. Schnürer
We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.
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We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson \cite[Problem 5.7]{Brothers86_OpenProblems}.
We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere after rescaling.
We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function $P$ and integral identities. In dimension $2$, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.
In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-regular and mean convex (but not area-minimizing), we can prove that the corresponding self-expanding hypersurfaces are smooth, embedded, and have positive mean curvature everywhere (see Theorem 1.1). As a result, for regular minimal but not area-minimizing cones we can give an affirmative answer to a problem arisen by Lawson …
A capillary problem for spacelike mean curvature flow in a cone of Minkowski space
本文研究了Minkowski空间中锥体的毛细问题，证明了在特定条件下，时空平均曲率流的抛物型边界值问题存在全局解，并趋于自相似的扩张解。
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.
Ancient mean curvature flow asymptotic to a minimal quadratic cone
本文证明n≥5时，渐近于O(n)×O(n)对称西蒙斯锥且位于其一侧的平均曲率流古解在抛物区域内有唯一渐近行为，平均凸性假设下则完全唯一，且为Hardt-Simon叶状结构的定常流。
Optimal regularity of optimal transport and degenerations of convex cones
本文通过仿射退化方法研究有界凸域间最优传输映射的边界正则性，提出稳定性准则与SL(n)模空间原理，并给出齐次最优传输映射存在性的几何判据。
Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on …
Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability
本文研究了常标量曲率凯勒锥度量的存在性，证明其与log K-能量的适切性及测地稳定性等价，并探讨了相关应用。
A partially overdetermined problem for $p$-Laplace equation in convex cones
该研究针对凸锥与光滑有界域外区域相交域中的p-拉普拉斯方程部分超定问题，建立容位势的存在性、正则性与渐近行为，借助P-函数、等周不等式及Heintze-Karcher型不等式，在正交相交假设下推导出刚性结果。
This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the very detailed asymptotic behaviour of constant scalar curvature Kahler (cscK) cone metrics, which leads to the reductivity of the automorphism group. Then we establish the linear theory for the Lichnerowicz operator, which immediately implies the openness of the path deforming the c…
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend the results on special Lagrangian submanifolds on $\mathbb C^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.
Given a smooth, symmetric, homogeneous of degree one function $f=f\left(λ_{1},\cdots,\,λ_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in $\mathbb{R}^{n+1}$, we show that under some suitable conditions on $f$ and the covariant derivatives of the second fundamental form of $\mathcal{C}$, there is at most one $f$ self-shrinker (i.e. an oriented hypersurface $Σ^n$ in $\mathbb{R}^{n+1}$ for which $f\left(κ_{1},\cdots,\,κ_{n}\right)+\frac{1}{2}X\cdot N=0$ holds, where $X$ is the position vector, $N$ is the unit nor…
We show that any smooth solution to the mean curvature flow equations coming out of a rotationally symmetric double cone is also rotationally symmetric.
Serrin's type Problems in convex cones in Riemannian manifolds
本文研究 Riemann 流形凸锥中的 Serrin 型超定问题，建立扭曲乘积与 Einstein 情形的刚性结果、肥皂泡定理及 Heintze–Karcher 不等式，并讨论 Euclid 锥上漂移 Laplace 算子的类似问题。
In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contain…
In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface $Γ\subset S^n$. The theory of isoparametic hypersurfaces is extremely rich and there are infinitely many distinct classes of examples, each with infinitely many members.
We use elementary methods to construct a minimal lamination of the interior of a positive cone in R3.
Ruled zero mean curvature surfaces in the three-dimensional light cone
该研究得到三维光锥中直纹零平均曲率曲面的完整分类，通过分析空间形式的测地线和螺旋运动发现螺旋面，并利用Weierstrass表示探讨其与悬链面的关系。
Björling problem for zero mean curvature surfaces in the three-dimensional light cone
本研究解决了三维光锥中零平均曲率曲面的Björling问题，并以此为基础构造且分类了所有旋转型零平均曲率曲面，为相关微分几何研究提供了新结果。
Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of the cone, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere as time tends to infinity.
We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.
We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-similar solutions, we give a priori estimates for the radially symmetric solutions and prove the global existence. Then we consider the homogenization limit as $\ve\to 0$, and use {\it the slowest self-similar solution} to characterize the solution, with error $O(1)\ve^{1/6}$, in s…
Bernstein-type Theorems for constant mean curvature surfaces in the three-dimensional light cone
本文在Gauss曲率下界条件下，证明了三维光锥上整个常均曲率图是horosphere或光锥上的球面。
Garding cones and positivity of curvature operators
本文研究Garding锥的包含关系，结合曲率算子正性，建立代数正性条件与黎曼流形几何的新联系，并揭示移位锥条件对拓扑的约束，如Betti数消失定理与球面空间形式刻画。
Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
本文证明了 Kähler–Ricci 收缩子的标量曲率仅依赖维数的一致界，并建立了非紧收缩子无穷远处唯一切锥与其极化 Fano 纤维化基的同胚关系。
Curvature growth and holomorphic splitting for Kähler metrics with cone singularities
本文研究锥角在1/2到1之间的Kähler度量，证明除子附近有界曲率导致法丛全纯分裂，证实Donaldson的猜想。
We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulae to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.
This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges towards a homothetically expanding hyperbolic solution.
Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $β\in (0,1)$. We show that in any positive co-homology class on $X$ there is a Kähler metric with cone angle $2πβ$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.
We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.
Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation
本文构造了一族平均曲率流，在 type I 和 type II 重标度下分别收敛到严格稳定的极小超锥及其 Hardt-Simon 叶状结构，并去除了 Velázquez 先前结果中的对称性条件。
Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the Kähler case. We also discuss some applications to complete Kähler manifolds with nonnegative bisectional curvature.
On uniform log $K$-stability for constant scalar curvature Kähler cone metrics
本文研究常标量曲率 Kähler 锥度量与 log $K$-稳定性的关系，证明其存在蕴含相应多稳定性与一致稳定性，并给出高次丰富除子上的存在性及相关性质。
We study the Kaehler metric given by the logarithm of a cubic form on its complexified index cone. Under mirror symmetry, this metric should asymptotically correspond to the Weil-Petersson metric. Using the theory of special Kaehler manifolds, a proof of a curvature formula for this metric is given.
We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.
We derive the equation of self-similar solutions to mean curvature flow based on the generalized Lawson-Osserman cone and prove the existence of self-expanders by modifying the theory of equilibria in the autonomous system. In particular, those self-expanders are unique if a local assumption is given.
We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperboles are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant $c$ from being the inner product between its tangent vector field and a fixed vect…
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).
This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegat…
We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $Σ$ immersed in $\mathcal{C}$ with free boundary in $\partial\mathcal{C}\setminus\{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape.
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
Foliation of null cones by surfaces of constant spacetime mean curvature near MOTS
研究通过流形技术证明在null锥体邻域内可由恒定时空中平均曲率面切分稳定MOTS，并提供构造指定时空中平均曲率面的方法。
We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity.
This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature con-ditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation:
Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone
该研究针对洛伦兹-闵可夫斯基时空中余维二、具平行平均曲率向量场的紧致类空子流形建立积分不等式，推导出刚性结论，完整刻画了光锥内这类子流形必为类空超平面中的全脐球面。
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we prove that local rigidity holds and obtain that the space of convex hyperbolic polyhedra with given combinatorial type is locally par…
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a…
The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.
We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a wea…
Nonnegative Ricci curvature, metric cones, and virtual abelianness
本文研究非负 Ricci 曲率开流形的基本群，在逃逸率非 1/2 且万有覆盖渐近为度量锥时证明其虚拟阿贝尔性，并在欧氏体积增长常数至少为 L 时给出指数界 C(n,L)。
In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in scalar curvature geometry saying that any (n-1)-dim. homology class of a compact manifold M^n with positive scalar curvature can be represented by a smooth (!) compact hypersurface that admits a positive scalar curvature metric.
Given a compact and H-convex subset $K$ of the Heisenberg group ${\mathbb H}$, with the origin $e$ in its interior, we are interested in finding a homogeneous H-convex function $f$ such that $f(e)=0$ and $f\bigl|_{\partial K}=1$; we will call this function $f$ the ${\mathbb H}$-cone-function of vertex $e$ and base $\partial K$. While the equivalent version of this problem in the Euclidean framework has an easy solution, in our context this investigation turns out to be quite entangled, and the problem can be unsolvable. The approach we follow makes use of an extension of the notion of convex f…
Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone
该研究针对欧氏空间原点中心的光滑凸锥内、与锥垂直的星形严格平均凸超曲面，分析速度为$(f(r)H)^{-1}$的逆曲率流，证明了$f$导数满足约束时流的长时间存在性及超曲面光滑收敛到球面片的结论。
We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence t…
Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds
本文利用扭曲Dirac算子的指标理论，证明奇异流形上数量曲率的Llarull型比较结果，并将Lipschitz刚性从偶数维推广到奇数维。
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