Venue

Conference Hall on 3rd Floor of Material Science Research Building (Section B&C), East Campus of USTC

Schedule

TBA

Titles & Abstracts

Shuli Chen (The University of Chicago)

Title: Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay

Abstract: We prove that if an orientable 3-manifold M admits a complete Riemannian metric whose scalar curvature is positive and has at most C-quadratic decay at infinity for some C > 2/3, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and S²×S¹ summands. Consequently, M carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant 2/3 is sharp, as demonstrated by metrics on R²×S¹. This improves a result of Balacheff, Gil Moreno de Mora Sarda, and Sabourau, and partially answers a conjecture of Gromov. The main tool is a new exhaustion result using μ-bubbles.

Shubham Dwivedi (University of Hamburg)

Title: Ricci-harmonic flow of G₂-structures

Abstract: We will talk about a new general flow of G₂-structures which we call the Ricci-harmonic flow. The flow is a natural coupling of the Ricci flow of underlying metrics and the isometric flow of G₂-structures but with explicit lower order terms. We will explain how to obtain the lower order terms and how the flow can be interpreted as an analog of the Ricci flow of metrics for G₂-structures. We will discuss various analytic and geometric results for the flow including short-time existence and uniqueness of solutions, regularity theory and long time existence results. Time permitting, we will also talk about explicit solutions and singularities of the flow. The talk is based on arXiv:2601.05210 and arXiv:2601.16832 (joint with Ragini Singhal).

Carlos Esparza (UC Berkeley)

Title: Existence and uniqueness of asymptotically conical shrinker Kähler-Ricci solitons

Abstract: We prove the uniqueness up to automorphisms of asymptotically conical Kähler-Ricci shrinkers on a fixed noncompact complex manifold M and derive obstructions to their existence. In particular, M needs to be a K-polystable polarized Fano fibration in the sense of Sun-Zhang. Joint work with Charles Cifarelli.

Hanbing Fang (Stony Brook University)

Title: Ricci flow limit space and its singular set

Abstract: Ricci flow is a powerful tool for studying the geometry and topology of smooth manifolds. In this talk, we focus on closed Ricci flows with a uniform lower bound on entropy. We first present a weak compactness theorem for the moduli space of such flows, each equipped with a natural spacetime distance, under pointed Gromov–Hausdorff convergence. The limiting object is referred to as a noncollapsed Ricci flow limit space. We then discuss its structure theory and analyze the singular sets. In particular, we describe a strong uniqueness theorem for the cylindrical tangent flow based on a Łojasiewicz-type inequality. From this, we can further establish that the cylindrical singular set is k-parabolic rectifiable. In dimension four, we prove that the singular set is 2-parabolic rectifiable, together with an integral estimate at the first singular time. This resolves Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows. This talk is based on joint work with Yu Li.

Siarhei Finski (École Polytechnique)

Title: Kodaira-Iitaka dimension and multiplicity: an analytic perspective

Abstract: We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.

Udhav Fowdar (University of Warsaw)

Title: The holonomy of the Obata connection on Joyce hypercomplex manifolds

Abstract: The Obata connection is the unique torsion free connection preserving a hypercomplex structure; it can be viewed as the hypercomplex analogue of the Levi-Civita connection on Riemannian manifolds. While a lot is known about Riemannian holonomy groups (including a full classification), very little is known about the Obata holonomy group. In this talk, I will present some new results exploring the Obata holonomy on compact Lie groups endowed with left invariant hypercomplex structures. The talk is based on a joint work with Beatrice Brienza, Giovanni Gentili and Luigi Vezzoni.

Nan Li (City University of New York)

Title: A Canonical Proof of Perelman's Stability Theorem

Abstract: Perelman's remarkable stability theorem states that if X is a compact n-dimensional Alexandrov space with curvature ≥ k, then for any ε >0, there exists δ = δ (X, ε)>0 such that for any n-dimensional Alexandrov space Y with curvature ≥ k (e.g. manifold with sec ≥ k), if Y is Gromov-Hausdorff close to X by a δ -approximation f: X → Y, then there is a homeomorphism g: X → Y which is ε-close to f.

We present a new and canonical proof of this result, which may help to improve Perelman’s Stability Theorem.

Sébastien Picard (University of British Columbia)

Title: Special Lagrangian submanifolds and circle collapse on K3

Abstract: We consider K3 surfaces degenerating to a three-dimensional affine base. We discuss how straight lines on the base lift to degenerating sequences of special Lagrangian two-spheres. The proof involves a gluing construction of compact special Lagrangians. This is joint work with Federico Trinca.

Jingzhou Sun (Shantou University)

Title: On the Bergman kernel of complex manifolds of constant holomorphic sectional curvature

Abstract: We will talk about our recent result on the Bergman kernel of polarized complex hyperbolic manifolds and polarized abelian varieties.

Jian Wang (University of Chinese Academy of Sciences)

Title: Toric exhaustion and Positive scalar curvature

Abstract: A 3-manifold is said to admit a toric exhaustion if it admits an exhaustion by solid tori. In this talk, we will discuss the existence of positive scalar curvature metrics on 3-manifolds admitting a toric exhaustion. As an application, we obtain a topological classification of irreducible 3-manifolds with positive scalar curvature and bounded geometry. 

Chengjian Yao (ShanghaiTech University)

Title: Closed G₂-structure with T³-symmetry and Hypersymplectic structures

Abstract: We use canonical decomposition of G₂-structure adapted to 3-dimensional subspaces to study closed or torsion-free G₂-structure with effective T³-symmetry. Closed G₂-structures are roughly classified into two types according to the orbits being non-isotropic or isotropic. The action is almost-free and the the orbit space is a good hypersymplectic orbifold with cyclic isotropic groups in the first type. The action is locally multi-Hamiltonian in the second type, and the open and dense subset of principal orbits is foliated by T³-invariant hypersymplectic manifolds. For complete torsion-free G₂-structures, we prove several Liouville type theorems under certain geometric assumptions. This is based on the joint work with Ziyi Zhou.  

Junsheng Zhang (New York University)

Title: Gromov-Hausdroff limits of immortal Kähler-Ricci flows

Abstract: We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian. This is based on joint work with Man-Chun Lee and Valentino Tosatti.