International Workshop on Frontiers in Geometry and Topology

Title:The rigidity of dimension estimate for holomorphic functions on Kähler manifolds

Abstract:In this talk, we will discuss the optimal rigidity of dimension estimate for holomorphic functions with polynomial growth on Kähler manifolds with non-negative holomorphic bisectional curvature. There is a specific gap between the largest and the second largest dimension. We also determine the optimal dimension that ensures the maximal volume growth which implies the manifold is biholomorphic to the complex Euclidean space. This is a joint work with Jie Deng, Zihang Hao and Jian Li.

Title:A singular dimension descent method for Geroch conjecture

Abstract:In this talk, I will introduce in detail the singular dimension descent method for Geroch conjecture, developed in the joint preprint arXiv:2606.20528 with Yuchen Bi from University of Freiburg.

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.

Title:Polyhedral CAT(0) metrics on open manifolds

Abstract:We give a near-complete topological characterization of open manifolds of dimension at least five that admit complete polyhedral CAT(0) metrics. We prove that they are pseudo-collarable and have perfectly semistable fundamental groups at infinity, and we establish a direct link to the arborescent structure of locally finite complexes. Joint work with Karim Adiprasito.

Title:Improvement of regularity for the pseudo Calabi flow

Abstract:Pseudo Calabi flow is a generalization of Kähler-Ricci flow to an arbitrary Kähler class, the critical points of which give cscK metrics. Many of the elliptic estimates which worked in the cscK case no longer work in the flow case. I will explain the differences in the elliptic and parabolic case, and will present an improvement of regularity result for the pseudo Calabi flow. This is joint work with Junhao Tian.

Title:Deformation rigidity for ℤ/2 eigensections

Abstract:Recently, a lot of attention has been drawn by so-called ℤ/2 harmonic functions, whose regularity is closely related to certain ℤ/2 eigensections. I will talk about a rigidity result for generic ℤ/2 eigensections. This is a joint project with Siqi He and Andries Salm.

Title:Index theorem and trace map of ℤ/2-harmonic spinors on 4-manifolds

Abstract:Let M be a 4-dimensional closed spin manifold and Σ be a cooriented 2-dimensional submanifold. In this talk, we will consider two elliptic boundary problems of Dirac operators defined on M ∖ Σ. One is defined by an APS (Atiyah-Patodi-Singer)-type boundary condition on the manifold with an edge metric. The other is defined by the deformation of ℤ/2-harmonic spinors. We will see how the indices of these elliptic boundary problems are related to their trace maps, as well as the relation between these trace maps. According to this observation, we can formulate the index formula for ℤ/2-harmonic spinors on 4-manifolds. This is joint work with Rafe Mazzeo and Andriy Haydys.

Title:Nonnegative Ricci Curvature and Uniformly Convex Boundary Forces Compactness

Abstract:We confirm a compactness conjecture of M. Li. If a complete Riemannian manifold has nonnegative Ricci curvature and uniformly convex boundary in the sense that the second fundamental form satisfies h ≥ 1, then we prove it is compact, and consequently has finite fundamental group. The proof uses monotone quantities constructed via positive proper harmonic functions with Neumann condition.

Title:From quasi-isometries to diffeomorphisms

Abstract:We will give geometric conditions on two manifolds X and Y in order that a quasi-isometry between them can be deformed into a diffeomorphism. We will also show several applications of our main result.

Title:Unknotting number and L-space satellite operators

Abstract:In a joint work with Daren Chen and Ian Zemke, we study the torsion order of Heegaard Floer homology under L-space satellite operators using a formula by Chen-Zemke-Zhou. By a result by Alishahi-Eftekhary, this leads to an unknotting number bound. The argument resembles the work of Hom-Lidman-Park which studies the torsion order of Heegaard Floer homology under cables using immersed curves. In this talk I will briefly review the tools and arguments used in both works.

Title:3-orbifolds and positive scalar curvature (Part 1)

Abstract:In these two talks, we discuss a classification result on 3-orbifolds with positive scalar curvature. Joint work with Ilaria Mondello and Thomas Richard.

Title:3-orbifolds and positive scalar curvature (Part 2)

Abstract:In these two talks, we discuss a classification result on 3-orbifolds with positive scalar curvature. Joint work with Ilaria Mondello and Thomas Richard.

Title:Adiabatic Limit Construction of Generalized ASD Instantons

Abstract:In the late 90s, Donaldson-Segal proposed the framework of instantons over higher dimensional manifolds for the purpose of studying metrics with special holonomy. In this talk, I will discuss how to apply the idea of adiabatic limits to construct such instantons over product manifolds. Meanwhile, I will also discuss some specific examples. This is joint work with Dylan Galt.

Title:The existence of taut foliations with zero Euler class

Abstract:It is known that every oriented plane field on a closed 3-manifold is homotopic to an integrable one. However, this no longer holds if one requires the foliation to be taut. This leads naturally to the question of which second cohomology classes can arise as the Euler classes of co-oriented taut foliations on a given 3-manifold M. When M is a rational homology sphere, the second cohomology group is finite, and the zero class plays a distinguished role. In this talk, we present infinitely many rational homology 3-spheres, including small Seifert fibred, hyperbolic, and toroidal examples, that admit co-oriented taut foliations but do not admit any with vanishing Euler class. We will also discuss the implications of these examples in the context of the L-space conjecture. This is joint work with Steve Boyer, Cameron Gordon and Duncan McCoy.

Title:Hypersymplectic structures and the geometry of closed G₂-structures

Abstract:Hypersymplectic structures appear naturally both in the study of symplectic Calabi-Yau manifolds in dimension 4, and in the study of G₂-geometry in dimension 7. For the former, they relate to a pure symplectic characterization of hyperkähler 4-manifold via Donaldson’s conjecture. For the latter, they are essential building blocks for closed G₂-structures with T³-symmetry. We will survey some results about hypersymplectic structures and closed G₂-structures with T³-symmetry, emphasizing the interplay of the geometries in the two respective dimensions.

Title:Gravitational instantons and harmonic maps

Abstract:It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane H², while in the Riemannian setting, 4-dimensional Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps and application includes a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large second Betti number b₂. Joint work with Song Sun.

Title:A positive mass theorem for continuous metrics

Abstract:Let g be a continuous metric on ℝ³ which is asymptotically flat in the sense that |g_ij(x) − δ_ij| = O(|x|^−τ) for some τ > 1/2. Further assume that g can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. For such a metric g, we define a synthetic ADM mass m(g) using harmonic functions. The harmonic mass m(g) coincides with the usual ADM mass whenever g is smooth and decays rapidly enough that the latter is defined. The harmonic mass can also be computed as a limit of the C⁰ local mass introduced by Burkhardt-Guim. Our main result is a positive mass theorem: the harmonic mass satisfies m(g) ≥ 0 and if m(g) = 0 then g is flat.

Title:cscK surfaces in algebraic families

Abstract:The remarkable theorem of Donaldson-Sun says that for non-collapsed polarized Kähler manifolds with bounded Ricci curvature, the Gromov-Hausdorff convergence can be realized as algebraic convergence inside a fixed Hilbert scheme. Based on this, Donaldson proved the Zariski openness of Kähler-Einstein locus inside a smooth family of Fano manifolds with discrete automorphism group. Both results rely on Hörmander’s L²-estimate, which requires a uniform Ricci lower bound. In this talk, I will discuss how to reach the same conclusion for non-collapsed polarized cscK surfaces, where uniform Ricci lower bound usually fails. Based on joint work with Junsheng Zhang.