Klaus Kroencke
We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M, g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold
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In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$.
Rigidity theorems for cone structures
该研究针对从常微分方程经典范构造得到的锥结构，证明以n维射影空间一般曲线或3维射影空间直纹曲面为模型的平凡锥结构是平坦的，并探讨了其在相关领域的应用。
New Inhomogeneous Nearly $\mathrm{G}_2$-metric on the Berger Space and a Sine Cone Desingularisation
本文在 Berger 空间上构造首个非齐次近 G2 度量，其锥具有 Spin(7) 全和乐，并通过计算机辅助证明存在性，同时用 C7 气泡实现正弦锥商群的局部去奇化。
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。
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-sing…
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…
We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature $κ\in \{-1,0,1\}$ and cone-angles $\leq π$. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first $L^2$-cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first $L^2$-cohomology group of the smooth part with coeffici…
Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces
本文证明具有锥奇点的闭球面之间的极小拉格朗日微分同胚必为等距映射，并将Liebmann刚性定理推广到常正高斯曲率闭曲面的分支浸入情形。
We show that the only metric measure space with the structure of an $N$-cone and with two-sided synthetic Ricci bounds is the Euclidean space ${\mathbb R}^{N+1}$ for $N$ integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the heat kernel in the $L^2$-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the $N$-dimensional standard sphere ${\mathbb S}^N$ as the unique minimizer of $$\int_X\int_X \cos d (x,y)\, m(d y)\,m(d x)$$ among all metri…
For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than $2π$. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies …
Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2π$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'
The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than π. We pr…
In \cite{CC1}, Cheeger-Colding considered manifolds with lower Ricci curvature bound and gave some almost rigidity results about warped products including almost metric cone rigidity and quantitative splitting theorem. As a generalization of manifolds with lower Ricci curvature bound, for metric measure spaces in $RCD(K, N)$, $1<N<\infty$, splitting theorem \cite{Gi13} and "volume cone implies metric cone" rigidity for balls and annuluses of a point \cite{PG} have been proved. In this paper we will generalize Cheeger-Colding's \cite{CC1} result about "almost volume cone implies almost metric c…
We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hyperbolic 3-cone-manifolds under the same angle assumption.
We show that an $n$-dimensional surface whose entropy is close to that of an $n$-dimensional plane is close in Hausdorff distance to some $n$-dimensional plane at every scale. Moreover we show that self-expanders of low entropy converge in the Hausdorff sense to their asymptotic cones.
In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.
We study the deformation of spherical conical metrics with at least some of the cone angles larger than $2π$. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction considered in recent works of Mazzeo and author.
Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions
本文研究标量曲率在奇数维奇异流形上的Lipschitz刚性，结合谱流论证和抽象锥算子分析，证明了奇数维Riemannian spin流形的刚性结果。
Rigidity and Morse index of free boundary minimal $T$-surfaces in the unit ball
本文证明平坦 $T$-锥到 $\mathbb{B}^n$ 的任何共形自由边界极小浸入在正交变换下与平坦 $T$-锥全等，从而统一了经典 Plateau 模型域的刚性定理。
We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on R^n.
Rigidity properties of holomorphic Legendrian singularities
本文在复解析范畴中研究接触流形的勒让德奇点，证明了约化切锥奇点与切锥的接触同胚双全纯等价性，以及正规勒让德奇点的形变刚性，方法基于无穷小接触同胚与自然线丛截面的联系。
Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splitting…
Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are taken toward showing that this holds also for flat cone surfaces, and it is shown that the lengths of closed geodesics are also enough to determine which of these three categories a geometric surface falls into. Techniques of Gromov, Bonahon, and Otal are explained and adapted, such…
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。
Rigidity and Classification of Legendrian Self-Shrinkers
本文对$\boldsymbol{\text{R}}^3$和$\boldsymbol{\text{R}}^5$中的勒让德自收缩子分类，证明类似Li-Wang的勒让德刚性定理，给出满足特定条件的勒让德自收缩子的结构结论。
We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2π$, i.e. contained in the interval $(0,2π)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A. Casson.
New rigidity theorem of Einstein manifolds and curvature operator of the second kind
本文通过Bochner技术证明了满足特定锥条件的高维爱因斯坦流形具有常曲率，并推广了相关研究结果。
We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.
New Curvature Operator of the Second Kind Conditions for Rigidity
本文通过第二类曲率算子建立Tachibana型刚性定理，证明维数≥5的完备Einstein流形在$C_n$-非负条件下具有常非负截面曲率，并推广至平移算子与对称空间分类。
Directional growth of coamenable normal subgroups: counterexamples and rigidity
研究高秩李群中余可安正规子群的增长刚性，发现方向性增长不守恒，但黎曼临界指数保持刚性，且对合不动点处增长指标一致。
Flexibility and rigidity of conformal embeddings in Lorentzian manifolds
该研究探讨洛伦兹流形中共形嵌入的柔性与刚性，证明闭可定向曲面的类空嵌入可被光滑共形嵌入$C^0$逼近，且特定商空间中可等距嵌入的负曲率度量集在泰希米勒空间中投影为相对紧集。
Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity
本文研究非负里奇曲率开流形的刚性，证明渐近锥包含欧氏空间时第一贝蒂数受维数差控制且等号情形流形平坦，并给出轨道增长阶的几何刚性及首个增长阶可变的例子。
Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations
该研究针对非均匀对角膨胀下的欧氏极小超曲面，通过对和非共振条件与降维论证证明其第二基本形式恒为零，还给出了相关刚性判据与生产函数应用。
Boundary Cases of the $J$-Equation: Divisorial Rigidity and a Global $C^0$ Estimate
该研究针对J方程稳定性条件的两类边界情形，通过分析不稳定素除子的例外族性质、修正nef估计及近似扭曲方程的全局$C^0$估计，构造出J方程的有势Bedford-Taylor解。
The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra, which deals with deformations that preserve face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volum…
We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic $G$-structure, corresponding to a reduction of the structure group of the tangent bundle. $S$ is characterized as the unique simply-connected compact complex manifold admitting such a $G$-structure which is at the same time integrable. To prove the deformation rigidity of $S$ it …
Time Separation and Scattering Rigidity for Analytic Lorentzian Manifolds
本文证明了实解析洛伦兹流形在时间分离函数和散射关系下的三类刚性结果，在无边界凸性等较弱假设下，实现了流形或子集在解析等距意义下的唯一确定。
Higher regularity for singular Kähler-Einstein metrics
本文研究奇异Kähler-Einstein度量在切锥处的收敛速率，证明了在Kähler势层面及通常度量意义下的多项式收敛性，并建立了具有极大体积增长的完备Calabi-Yau度量的刚性结果。
This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angles) can be turned into effective control on the deformations that do exist. This leads to precise analytic and geometric versions of the idea that hyperbolic structures with short geodesics are close to hyperbolic structures with cusps. The paper also outlines a new harmonic defo…
Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces
本文在RCD(0,N)度量测度锥中建立周长极小化集合的单调性公式及刚性陈述，并由此得到非塌缩RCD空间中奇异分层的尖锐Hausdorff维数估计，以及非负Ricci曲率流形中全局周长极小化子blow-down锥的存在性。
Donaldson-Sun Theory in the Conic Case
将Donaldson-Sun度量切锥理论推广到边界系数为有理数有限子集的锥Kähler-Einstein对的非塌缩Gromov-Hausdorff极限，证明对数度量切锥的唯一性，关联稳定退化机制，构造反例表明Kähler-Einstein假设对切锥刚性的必要性。
Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature
本文证明了具有非负 Ricci 曲率和 Euclidean 体积增长的测度 Finsler 流形的尖锐等周不等式，并在附加假设下证明了其刚性结果，进而推导出 Euclidean 空间中锥的加权各向异性等周不等式在不可逆情形下的刚性。
Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
本文研究了满足 RCD(0,N) 条件的度量测度空间中夹紧板主频率的精细性质，在 N 接近 2 或 3 时提出了包含渐近体积比的新等周不等式，并证明了其锐度、基于锥结构的刚性及稳定性，结果对黎曼流形亦属全新。
We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $π$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure.
We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $ν$ to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates $ν< -4$ in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicit…
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