Jianchun Chu
We prove a $C^{1,1}$ estimate for solutions of a class of fully nonlinear equations introduced by Chen-He. As an application, we prove the $C^{1,1}$ regularity of geodesics in the space of volume forms.
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Regularity for the Geodesic X-Ray Transform in Nonsmooth Geometry
本文研究非光滑简单流形上测地线X射线变换的正则性，通过符号平滑和低正则伪微分算子映射性质，改进了$L^p$函数上变换的单射性结果。
Regularity of a Geodesic equation in the space of mixed Volume Forms on Hermitian Manifolds
研究Hermitian流形上混合体积形式空间中测地线方程的正则性，证明在椭圆性条件下该退化方程存在C^{1,1}解，并构造显式下解。
We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Ampère equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.
A Partial $ε$-Regularity Theorem for Integral Stationary Geodesic Networks in $\mathbb{R}^n$
研究\(\mathbb{R}^n\)中整数重数平稳测地线网络序列，通过证明部分\(\epsilon\)-正则性定理，得出其收敛到重数为\(k\)直线的结论。
This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Ampére type eqution on the Kähler cone; secondly, we obtain a priori etimates for the above equation. These a priori estimates guarantee the existence and uniqueness of $C^{2}_{w}$ geodesic for any two points in the space of Sasakian metrics. We also give some geometric applications of the above estimates in the end of this paper.
We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähler metrics associated to a test configuration, as well as the local $C^{1,1}$ regularity of quasi-psh envelopes in nef and big classes away from the non-Kähler locus.
It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,α}, 0<α<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.
We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value problem for geodesics for locally Lipschitz continuous metrics and generalize recent results on existence, regularity and uniqueness of solutions in the sense of Filippov.
Regularity of the geodesic flow on submanifolds
本文证明了$\mathbb{R}^n$中$C^k$子流形的测地流和指数映射具有$C^{k-1}$类正则性。
On the regularity of geodesic rays associated to test configurations
利用奇点的奇点的分解，对X上的正线丛L的任意检验位形构造了C^{1，1}类测地射线。该构造简化为寻找相应的Monge-Ampere方程的子解。在几何上，这是通过使用一个积极的线丛的分辨率，这是平凡的例外除数之外。
We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand side.
We prove a so called $κ$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $κ$ non-collapsing property for Ricci flow. These two results together imply volume doubling property for Ricci flow without assuming Ricci curvature lower bound.
In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves between two points, in connection with infinite dimensional techniques in Morse Theory.
We study asymptotic topological regularity of CAT(0) spaces. We prove that if a purely n-dimensional, proper, geodesically complete CAT(0) space has small volume growth, then it is homeomorphic to the n-dimensional Euclidean space. We also discuss asymptotic geometry of proper, geodesically complete CAT(0) spaces of small volume growth.
On the affine parametrization of null geodesics in low regularity
本文针对低正则性时空与洛伦兹长度空间中类光测地线缺乏标准参数化的问题，讨论通过类时测地线极限定义仿参数化的方法，指出其存在非唯一结果的缺陷，并探索利用类时测地线构造彭罗斯型奇点定理的可能性。
We prove the smoothness of abnormal minimizers of subriemannian manifolds of step 3 with a nilpotent basis. We prove that rank 2 Carnot groups of step 4 admit no strictly abnormal minimizers. For any subriemannian manifolds of step less than 7, we show all abnormal minimizers have no corner type singularities, which partly generalize the main result of Leonardi-Monti.
Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford-Taylor type envelope techniques developed in the authors previous work [ber2], that Chen's weak geodesic connecting any two elements in H are C^{1,1}-smooth, i.e. the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacins due to Chen. This also gives a partial generalization of Blocki's refinement of Che…
In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic cu…
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain $Ω$ in $\mathbb R^n$ with the Kobayashi distance of the tube domain over the domain $Ω$ is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tu…
Regularity of volume-minimizing flows on 3-manifolds
在这篇文章中，我们证明了，对于任何紧的3-流形，有一个$C^{1}$体积最小的一维叶理。更一般地说，我们证明了存在的质量最小化的可求长部分的球面丛没有孤立的“极点”的基础流形。这个相同的分析被用来表明，由于Sharon Pedersen，$S^{2n+1}$的单位切丛的潜在体积最小化的可求正截面（可求正叶理）的例子实际上不是体积最小化。
Noncritical maps on geodesically complete spaces with curvature bounded above
本文研究了曲率有上界的测地完备空间上距离映射的正则性，证明了其局部为Hurewicz纤维化，并作为推论得到了测地完备CAT(1)空间的球面定理。
Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds
研究实解析子黎曼流形中严格异常极小测地线的正则性和分支特性，通过构造例子解决长期未解问题，并在 Carnot 组中证明非光滑和分支极小测地线的存在。
Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.
We prove that a reversible Berwaldian Finsler structure $\left(M,F\right)$ with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.
Geometric $C^1$-stability of geodesic spheres under volume-preserving mean curvature flow
该研究在单连通空间形式中，通过定量Alexandrov反射论证，证明保体积平均曲率流下测地球面的几何C¹稳定性，得到长时间存在性、最终凸性及光滑指数收敛结果。
S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its perturbed equation and prove that the space of volume forms is an infinite dimensional non-positively curved metric space in the sense of Alexandrov.
4-dimensional Space forms as determined by the volumes of small geodesic balls
本文基于张量微积分与经典定理而非拓扑刻画，证明了小测地球体积能确定4维流形是否为空间形式，扩展了Gray-Vanhecke猜想的相关结论。
Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane.
Integral Geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S^2 times S^2
我们证明了在Hamilton变形下，方程$S^{1} \times S^{1}$在$S^{2} \times S^{2}$中的乘积是全局体积极小化的.
We prove that the geodesic equation for any semi-Riemannian metric of regularity $C^{0,1}$ possesses $C^1$-solutions in the sense of Filippov.
Generic regularity of isoperimetric regions in dimension eight
本文在八维闭黎曼流形中证明等周区域的通用正则性，表明在度量与体积的通用选取下，等周区域边界光滑且非退化。
In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), rega…
An $ε$-regularity theorem for Perelman's reduced volume
本文研究Ricci流中Perelman约化体积的正则性，证明当其接近1时基点曲率半径具有下界。
The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped product. In particular, we present an obstruction for the existence of such a class of manifolds.
The Shortest Geodesic Flower on a Manifold with Locally Convex Ends and Finite Volume
研究了具有局部凸端和有限体积的流形上存在非平凡测地网的性质，证明了其顶点数、边数和总长度的上界。
We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the structure $M_{\rm geod}$ the unique structure with minimum convex core volume.
Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.
In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson \cite{MR2975584} to the case of Kähler manifolds with boundary; moreover we introduce a subspace $\mathcal H_C$ of $\mathcal H_\b$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^…
Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa--Holm equations are well-studied examples.A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description.Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. Ther…
Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one
本文证明含余维数一孤立闭全测地子流形的非正曲率闭流形具有正单纯体积，并推出非正弯曲解析流形在万有覆叠含余维数一平坦时的二分结论。
An improvement of regularity result for pseudo Calabi flow
本文改进了伪Calabi流的正则性结果，证明初始数据体积形式接近光滑形式时流立即光滑，并展示初始数据接近cscK度量时流在正时间存在。同时证明Fano流形上伪Calabi流的正则性和长期存在性结果。
Generalized $\Cal{L}$-geodesic and monotonicity of the generalized reduced volume in the Ricci flow
假设$M$是一个完备的n维流形$n\ge 2$，其度量$\bar{g}_{ij}(x,t)$由$M\times (0,T)$中的里奇流$\partial_t \bar{g}_{ij}=-2\bar{R}_{ij}$演化。对于任何$0<p<1$，$(p_0,t_0)\in M\times (0,T)$，$q\in M$，我们定义$\Cal{L}_p$-$p_0$和$q$之间的长度，$\Cal{L}_p$ -测地线，广义约化距离$l_p$和广义约化体积$\widetilde{V}_p(τ)$，$τ=t_0-t$，对应于时间$t_0$点$p_0$处的$\Cal{L}_p$ -测地线.在$M\times (0,t_0)$上的条件$\bar{R}_{ij}\ge -c_1\bar{g}_{ij}$下，对于某个常数$c_1>0$，我们将证明存在一条$\Cal{L}_p$ -测地线，它使得对于任何$\barτ>0$，$p_0$和$q$之间的$\Cal{L}_p(q,\barτ)$ -长度最小化.这一结果的情况$p=1/2$是证明和使用了很多次，但没有证明它是在佩雷尔曼的文件里奇流。我的结果是新的和答案肯定存在这样的$\Cal{L}$ -测地线极小的$L_p(q,τ)$ -长度这是至关重要的证明许多结果在佩雷尔曼的文件里奇流。我们还得到了广义$\Cal{L}_p$ -测地线和广义约化体积的许多其他…
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending…
Asymptotic geometric regularity of CAT(0) spaces
该研究证明了CAT(0)空间在特定条件下与欧几里得空间的双Lipschitz同胚性，并应用此结果得出CAT(1)空间的同胚结论。
Totally geodesic subgroups of diffeomorphisms
考虑右不变的$L^2$ -度量，我们确定了精确保体积微分同胚群是保体积微分同胚群的全测地子群的黎曼流形。在Kähler情况下，同样的事情也发生在作为辛同态群的子群的哈密顿同态子群上。这些是具有足够大的李代数以检测对称2-张量场的消失的全测地子群的特殊情况。
The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{pol}$} and the {\em weak polynomial entropy $h_{pol}^*$}. Both are infinite when the topological entropy is positive and they satisfy $h_{pol}^*\leq h_{pol}$. We first prove that the growth rate of the volume of balls is bounded above by means of the strong polynomial entropy and w…
We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.
Real-analytic geodesics in the Mabuchi space of Kähler metrics and quantization
本文研究 Kähler 度量 Mabuchi 空间中测地线的量子化逼近，证明实解析初值在短时间内量子化 Bergman 测地线收敛于 Mabuchi 测地线，并指出边值问题通常无解。
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-Hölder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.
We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with curvature-like invariants of the dynamics, appear in the expansion of the volume at regular points of the exponential map. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of geometric structures, including all sub-Riem…
We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geodesic flows, these lower bounds are: (i) For any Riemannian metric on M, any pair of non-conjugate points p,q in M, and every component C of the space of paths from p to q, the number of geodesics in C of length at most T grows at least like e^{\sqrt T}. (ii) The exponent of the vo…
Properness and finiteness of totally geodesic submanifolds in the convex core
本文研究几何有限秩一局部对称流形凸核内的全测地子流形，证明其凸核内维数≥2的极大全测地子流形正常浸入、体积有限且数量有限，结合已有工作推导得出含无穷多该类子流形的对应流形必为算术流形。
Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
针对闭双曲3-流形，研究其本原简单闭测地线数目随长度的增长规律，得出长度不超过L的该类测地线数目为e^((2+o(1))L)的结论
Geodesic simplices of pseudo-hyperbolic space
本文通过上同调解释伪双曲空间测地单纯形，并给出了其体积有限的充要条件，推导出二维双曲空间中理想测地多面体具有有限体积。
We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with geometrically finite limit. This allows us to show that the renormalized volume attains its minimum (in terms of the conformal class at $\partial M = S$) at the geodesic class, the conformal class for which the boundary of the convex core is totally geodesic.
Given a compact Kähler manifold $(X,ω_0)$ let $\mathcal H_{0}$ be the set of Kähler forms cohomologous to $ω_0$. As observed by Mabuchi \cite{m}, this space has the structure of an infinite dimensional Riemannian manifold, if one identifies it with a totally geodesic subspace of $\mathcal H$, the set of Kähler potentials of $ω_0$. Following Donaldson's research program, existence and regularity of geodesics in this space is of fundamental interest. In this paper, supposing enough regularity of a geodesic $u:[0,1]\to \mathcal H$, connecting $u_0 \in \mathcal H$ with $u_1 \in \mathcal H$, we est…
Let $F : H^q \to H^q$ be a $C^k$-map between Sobolev spaces, either on $\mathbb R^d$ or on a compact manifold. We show that equivariance of $F$ under the diffeomorphism group allows to trade regularity of $F$ as a nonlinear map for regularity in the image space: for $0 \leq l \leq k$, the map $F: H^{q+l} \to H^{q+l}$ is well-defined and of class $C^{k-l}$. This result is used to study the regularity of the geodesic boundary value problem for Sobolev metrics on the diffeomorphism group and the space of curves.
In this paper, we show that for a Poincaré-Einstein manifold $(X^{n+1},g_+)$ with conformal infinity $(M,[\hat{g}])$ of nonnegative Yamabe type, the fractional Yamabe constants of the boundary provide lower bounds for the relative volume. More explicitly, for any $γ\in (0,1)$, $$ \left(\frac{Y_{2γ} (M,[\hat{g}])}{Y_{2γ} (\mathbb{S}^n , [g_{\mathbb{S}}])}\right)^{\frac{n}{2γ}} \leq \frac{V(Γ_t(p),g_+)}{V(Γ_t(0),g_{\mathbb{H}})} \leq \frac{V( B_t(p),g_+)}{V( B_t(0), g_{\mathbb{H}})} \leq 1,\quad 0<t<\infty, $$ where $B_t(p)$, $Γ_t(p)$ are the the geodesic ball and geodesic sphere of radius $t$ i…
Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
本文在积分体积亏损界下证明非坍缩Ricci极限空间的度量奇异集余维数至少为四，并具有sigma有限测度，验证了余维四正则性猜想的一个特例，且该界是尖锐的。
On the singular set of the geodesic flow on nonpositively curved surfaces
该研究证明非正曲率曲面上测地流奇异轨迹的特征函数属于特定几何各向异性 Sobolev 空间，由此推导测地流非遍历的条件，证明依赖半经典分析与抛物流新径向估计。
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