Beijing Institute of Mathematical Sciences and Applications

A harmonic map heat flow admits a global (long-time) solution if the target manifold has nonpositive sectional curvature, by the classical theorem of Eells and Sampson. However, for general target manifolds without curvature assumptions, the flow may develop a singularity in finite time. In this talk, we focus on such finite-time blow-up and its characterization. At the maximal finite time, it is known that the "limsup" of the critical norm of the gradient of the map diverges to infinity. Recently, in joint work with Hideyuki Miura and Jin Takahashi, we proved that the "liminf" also diverges to infinity. I will outline the proof of this result.

It is well-known since Lichnerowicz that the index theory of Dirac operators plays an important role in the study of scalar curvature on spin manifolds. The spectral flow is an odd-dimensional counterpart of the Fredholm index. We will discuss how the spectral flow can be applied to solving questions related to scalar curvature, including the long neck problem, band width problem, quantitative estimate, in the odd-dimensional case.

Hultgren proved that the existence of coupled Kähler–Einstein (cKE) metrics on toric Fano manifolds can be characterized in terms of the barycenters of collections of associated polytopes. He also constructed an example of a toric Fano fourfold admitting a two-coupled KE metric but no ordinary KE metric. In this talk, we present higher-dimensional generalizations of his example, which yields a family of projective bundles that admit cKE metrics but not KE metrics. Moreover, we show that no such example exists among toric Fano threefolds for any k-coupled KE metric. This is joint work with Y. Hashimoto.

In this talk, we will recall some examples related to Ricci limit spaces: (1). For any $n\ge 3$, there exists an n-dimensional Ricci limit space has no open subset which is topologically a manifold. This generalizes a result of Hupp-Naber-Wang. As a corollary, our example provides a collapsed sequence of boundary free manifolds whose limit has a dense boundary with infinitely many connected components. (2). For any $n\ge 4$, there exists a sequence of n-dimensional tori with Ricci lower bound that converges to a singular space. This answers a question posted by Petrunin and Brue-Naber-Semola. In the 4-dimensional case, we prove that the Gromov-Hausdorff limit of tori with a two-sided Ricci bound and a diameter bound is always a topological torus. (3). For any discrete $\Gamma\lt O(4)$, there exist a small constant $\epsilon＞0$ and a complete Riemannian 4-manifold $(M,g)$ with nonnegative Ricci curvature, asymptotic to $C(S_{\epsilon}^3/\Gamma)$. This answers a question posed by Brue-Pigati-Semola.