abstracts trends-2024

We show that there exists a quantity, depending only on C^0 data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the C^0 sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the C^0 mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the C^0 local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. The PIC condition is preserved under the Ricci flow, and the construction of a flow with surgeries was used to give a complete classification of PIC manifolds in dimension 4 and in sufficiently high dimensions. There is a completely different approach to the study of PIC manifolds using minimal surfaces. In this talk we will give an overview of these two approaches, and discuss recent results on largeness properties of stable surfaces in PIC manifolds.

By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.

We will discuss a new approach to flows with surgery, which in particular applies in the setting of mean-convex surfaces with free boundary. As an application (joint with Dan Ketover) we establish the existence of multiple free boundary minimal disks in convex balls.

I will discuss various existence and uniqueness results for a class of curvature problems. In particular, I discuss kinematic curvature problems related to area measures and curvature measures.

In this talk, we will discuss the existence of Ricci flow from complete non-compact manifolds with possibly unbounded curvature. We will focus on manifolds with pinched non-negative 1-isotropic curvature and discuss its application to Hamilton-Lott pinching conjecture. This is based on joint work with Topping, Chan and Peachey.

In this talk we survey some known examples of ancient solutions of Ricci flow which are not Ricci solitons. We will emphasize the disparity in the current knowledge between the compact examples and noncompact examples.

Steady Ricci Solitons may arise as singularity models of the Ricci flow and they are essential to the singularity analysis in order to achieve any topological applications using surgeries, particularly in dimension 4. In this talk, we shall present some results on four-dimensional noncollapsed steady gradient Ricci solitons in some recent joint works. We first classify the tangent flows at infinity of noncollapsed steady solitons in dimension four. With the classification, we give a classification of Kaehler steady solitons. Under the assumption of positive curvature, we present some recent results on rough geometric asymptotics and precise analytic asymptotics.

This talk will survey several results regarding four-dimensional complete noncompact Ricci solitons, including their asymptotic structure at infinity, curvature estimates, and volume growth estimates. Several open problems will be presented.

Kodaira’s vanishing theorem is proved via Bochner formula applied to the square norms of harmonic forms. I shall explain a new approach via the maximum principle applied the Whitney’s co-mass.

The Hull-Strominger system arose in superstring theory, and is a subject of intense mathematical interest due to its connection to uniformization problems in complex geometry. I will discuss a mild reformulation of this system, and show that the tools of pluriclosed flow/generalized Ricci flow can be used to construct solutions to this system. This point of view leads to a proof of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system. Joint work with M. Garcia-Fernandez and R. Gonzalez-Molina.