Note on geodesic rays tamed by simple test configurations

On the regularity of geodesic rays associated to test configurations

Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor.

Test configurations, large deviations and geodesic rays on toric varieties

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets. A novelty in the analysis is the connection between Bergman metrics, Bergman kernels and the theory of large deviations.

$C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays

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.

Geodesic rays and stability in the cscK problem

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the $\mathcal{J}^{K_X}$-stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Ou…

Constant Mean Curvature Hypersurfaces Condensing to Geodesic Segments and Rays in Riemannian Manifolds

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out.

Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.

大上同调类中的有限能量测地射线

Finite Energy Geodesic Rays in Big Cohomology Classes

本文研究大上同调类中有限 p-能量测地射线，证明 ℰ^p 空间的 Busemann 凸性并构造完备弦度量，还借助 Ross-Witt Nyström 对应给出测地射线刻画。

Weak geodesic rays in the space of Kähler potentials and the class E(X,ω)

Suppose (X,ω) is a compact Kähler manifold. In the present work we propose a simple construction for weak geodesic rays in the space of Kähler metrics that seems to be tied together with properties of the class E(X,ω). As an application of our construction, we prove a characterization of E(X,ω) in terms of envelopes.

Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds

Suppose $(X,J,ω)$ is a Fano manifold and $t \to r_t$ is a diverging Kähler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's $\mathcal F$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fields this proves the equivalence between geodesic stability of the $\mathcal F$-functional and existence of Kähler-Einstein metrics. We also explore applications of our construction to Tian's $α$-invariant.

模型位势的非唯一性与弱测地线射线

Nonuniqueness of Model Potentials and Weak Geodesic Rays

该研究在复射影直线上揭示两种非唯一性现象：不同归一化正质量模型位势可对应相同非多重极蒙日-安培测度且相关局部性质一致，同时构造出零接触集逐点匹配、逆勒让德变换为同初始切向量的不同有界弱测地线射线的极大测试曲线，丰富了复几何中相关对象的结构认知。

Minimal rays on surfaces of genus greater than one

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the Busemann functions with these directions are unique up to adding constants. Moreover, using a kind of weak KAM theory, we show that for almost all types of minimal geodesics in the sense of Morse, there is precisely one minimal geodesic of this type.

Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in a Kähler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional analogue, where we show that t…

Teichmüller空间上的射线结构

Ray structures on Teichmüller Space

本文研究了Teichmüller空间中射线结构的几何过渡，通过能量最小化约束选择唯一测地线，并扩展到有边界表面的伸缩映射问题。

具有$T$-对称性的Kähler度量空间中的测地射线

Geodesic rays in the space of Kähler metrics with $T$-symmetry

本文在具有环面作用的Kähler流形上构造奇异混合极化和一族相容复结构，证明对应度量路径为完备测地射线，且极化族收敛于混合极化。

From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this res…

Conformal limits of grafting and Teichmüller rays and their asymptoticity

We show that any grafting ray in Teichmüller space is (strongly) asymptotic to some Teichmüller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a grafting ray, the proof involves a Teichmüller ray with a conformally equivalent limit, and building quasiconformal maps of low dilatation between the surfaces along the rays. Our preceding work had proved the result for rays determined by an arational lamination or a multicurve, and the unified approach here gives an alternative proof of the former case.

On K-polystability of cscK manifolds with transcendental cohomology class

In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature Kähler (cscK) metric implies 'geodesic K-polystability', in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen-Tang we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds. Hence our main result recovers a possibly stronger version of results of Berman-Darvas-Lu in this …

Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds

Let $σ$ be the scattering relation on a compact Riemannian manifold $M$ with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let $\ell$ be the length of that geodesic ray. We study the question of whether the metric $g$ is uniquely determined, up to an isometry, by knowledge of $σ$ and $\ell$ restricted on some subset $D$. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from $D$ covers $T^*M$; and that those geodes…

环面流形的 Mabuchi 射线、测试构型与量子化

Mabuchi rays, test configurations and quantization for toric manifolds

本文研究辛环面流形上与环面测试构型关联的 Mabuchi 射线，证明其在无穷测地时间处收敛到混合极化，并刻画了相应极限量子化的分解结构，以 $S^2$ 为例推广到高维情形。

On the space of light rays of a space-time and a reconstruction theorem by Low

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical properties of the space of light rays, i.e., unparametrized time-oriented null geodesics, $\mathcal{N}$. In the analysis of the reconstruction problem it becomes instrumental the structure of the space of skies, i.e., of congruences of light rays. It will be shown that the space…