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Gang Tian

Publications and source records attributed to Gang Tian.

At least 19 recordsLinked to original sources
arXiv2026

All-Genus Large-Degree Asymptotics for Gromov--Witten Invariants of the Projective Plane

This is the first part of a series of papers on the large-degree asymptotics of Gromov--Witten invariants. In this paper, we prove complete large-degree asymptotic expansions, at every fixed genus, for the primary Gromov--Witten invariants of the complex projective plane. The proof uses singularity analysis to transfer the local expansions of generating functions at their dominant singularities to asymptotic expansions of Gromov--Witten invariants. The genus-zero asymptotic expansion is obtained from an analysis of the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equation. The higher-genus cases are obtained from the Givental--Teleman reconstruction theorem for semisimple cohomological field theories and from the graph-sum formula for the $R$-matrix action.

math.AG↗
arXiv2025

Compactification of metric moduli space of $K3$ surfaces

We prove a conjecture of Odaka--Oshima, which says that there is an algebraic description of the Gromov--Hausdorff compactification of all unit-diameter hyperk\"ahler metrics on K3 surfaces. As a corollary, we obtain a classification of the Gromov--Hausdorff limits of those hyperk\"ahler K3 surfaces with a fixed complex structure or with a fixed polarization.

math.DG↗
arXiv2025

Laplace comparison on K\"ahler Ricci flow and convergence

We first prove a uniform integral Laplace comparison result for the K\"ahler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of K\"ahler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.

math.DG↗
arXiv2025

The biharmonic hypersurface flow and the Willmore flow in higher dimensions

The biharmonic flow of hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb {R}^{n+1}$ for $n\geq 2$ is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon-Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem in \cite{BWW} on the biharmonic hypersurface flow for $n=4$. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions.

math.DG↗
arXiv2025

Stability thresholds for big classes

In 1987, the $\alpha$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $\delta$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $\tau\in[0,1]$. The special degenerate (collapsing) case $\tau=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.

math.DG↗
arXiv2025

A geometric characterization of potential Navier-Stokes singularities

For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near a potential singularity, the directions of vorticity cannot avoid any great circle on the unit sphere. Our method, based on the control of local vorticity fluxes, is inspired by the classical Kelvin-Helmholtz law for ideal fluids and the Type I regularity theory for axisymmetric Navier-Stokes solutions.

math.AP↗
arXiv2024

Global solutions to the Euler-Coriolis system

We prove the global well-posedness and scattering for the 3D incompressible Euler-Coriolis system with sufficiently small, regular and suitably localized initial data. Equivalently, we obtain the asymptotic stability for "rigid body" rotational solutions to the pure Euler equations. This extends the recent work of Guo, Pausader and Widmayer to the general non-axisymmetric setting.

math.AP↗
arXiv2023

Finite time singularities of the Kähler-Ricci flow

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

math.DG↗
arXiv2023

Geometric regularity of blow-up limits of the Kähler-Ricci flow

We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-$W_1$ distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than $4$.

math.DG↗
arXiv2022

Horosymmetric limits of Kähler-Ricci flow on Fano $G$-manifolds

In this paper, we prove that on a Fano $\mathbf G$-manifold $(M,J)$, the Gromov-Hausdorff limit of Kähler-Ricci flow with initial metric in $2πc_1(M)$ must be a $\mathbb Q$-Fano horosymmetric variety $M_\infty$, which admits a singular Kähler-Ricci soliton. Moreover, $M_\infty$ is a limit of $\mathbb C^*$-degeneration of $M$ induced by an element in the Lie algebra of Cartan torus of $\mathbf G$. A similar result can be also proved for Kähler-Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the type II singularity of Kähler-Ricci flows on Fano $\mathbf G$-manifolds to Fano horosymmetric manifolds.

math.DG↗
arXiv2022

A Note On K\"ahler-Ricci Flow on Fano Threefolds

In this note, we show that the solution of K\"ahler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits K\"ahler-Ricci soliton and the Gromov-Hausdorff limit of the K\"ahler-Ricci flow must be a singular $\mathbb{Q}$-Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which K\"ahler-Ricci flow develops type II singularity.

math.DG↗
arXiv2022

Principal minors of Gaussian orthogonal ensemble

In this paper, we study the extremal process of the maxima of all the largest eigenvalues of principal minors of the classical Gaussian orthogonal ensemble (GOE). We prove that the fluctuation of the maxima is given by the Gumbel distribution in the limit. We also derive the limiting joint distribution of the maxima and the corresponding eigenvector, which implies that these two random variables are asymptotically independent.

math.PR↗
arXiv2022

Kähler stability of symplectic forms

Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a Kähler form remains Kähler.

math.DG↗
arXiv2021

Singular limits of Kähler-Ricci flow on Fano $G$-manifolds

In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of $\mathrm{SO}_4(\mathbb{C})$ and one Fano compactification of $\mathrm{Sp}_4(\mathbb{C})$, on which the Kähler-Ricci flow will develop singularities of type II. To the authors' knowledge, these are the first examples of Ricci flow with singularities of type II on Fano manifolds in the literature.

math.DG↗