arXiv research

Solves modified conjecture for Fano manifolds using Ding stability.

Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao …

Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…

In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations (Theorem 1.3). This extends the known result obtained by others (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. We also sho…

New stability criterion for Fano manifolds using anticanonically balanced metrics.

In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In …

New stability criteria for Fano varieties using generalized b-divisors.

We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existe…

New approach finds analytic interpretation of algebraic invariants for balanced metrics.

We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano …

We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the Kähler-Ricci soliton modulo automorphisms. In our results, we do not assume that the vanishing of the higher order modified Futaki invariants introduced by Berman-Nyström…

We compute the Hessian of quantized Ding functionals and give an elementary proof for the convexity of quantized Ding functionals along Bergman geodesics from the view point of projective geometry. We study also the asymptotic behavior of the Hessian using the Berezin-Toeplitz quantization.

It came to my attention after posting this paper that Yu Ding has proved the same result before. I would like to apologize to Yu Ding for the appearance of this paper.

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 fi…

The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.

Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

In this paper, we study Mabuchi metrics on Fano manifolds. We prove that Mabuchi metrics exist if the modified Ding functional is proper modulo a reductive subgroup of its automorphism group. On the other hand, the inverse that Mabuchi metrics implies the properness is obtained by using Darvas-Rubinstein's properness p…

Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.

New proof given for a functional's minimum condition.

K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups …

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 πλc_1(X)$ for $λ=\pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with …

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by $J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8π\int_M udv_g-8π\log\int_Mhe^udv_g,$ where $h$ is a positive smooth function on $M$. In an inspiring work (…

Eigenvalue estimate for shrinkers in mean curvature flow.

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…

Researchers introduce new energies to study constant scalar curvature metrics.

We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.

Analyzes Kähler-Einstein metrics on families of Fano varieties.

We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding-Geiges-Stipsicz for computing the d3-invarian…

We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kah…

The study pinches self-shrinking hypersurfaces in Euclidean space.

The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Eule…

In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover,…

The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted $Ding$-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth p…

We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtaine…

It is our purpose to study complete self-shrinkers in Euclidean space. First of all, we show some examples of complete self-shrinkers without polynomial volume growth. By making use of the generalized maximum principle for $\mathcal{L}$-operator, we give a complete classification for 2-dimensional complete self-shrinke…

The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positiv…

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we sh…

Study on compact Kähler surfaces for sign-changing curvatures.

Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.

In this paper, we introduce the "coupled Ricci iteration", a dynamical system related to the Ricci operator and twisted Kähler-Einstein metrics as an approach to the study of coupled Kähler-Einstein (CKE) metrics. For negative first Chern class, we prove the smooth convergence of the iteration. For positive first Chern…

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bo…

Efficiently matches random graphs with inhomogeneous edge probabilities.