Résumés 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.

In this talk, we first introduce geometric quantities for capillary hypersurfaces, and then establish isoperimetric type inequalities, Alexandrov-Fenchel inequalities, by studying corresponding geometric flows. We will also talk about a related Minkowski type problem and other related problems. The talk is based on a joint work with Chao Xia and other collaborators.

In this talk, we will present some joint work with Ovidiu Munteanu concerning a volumetric Minkowski inequality. The classical Minkowski inequality says that the surface area of a convex compact domain in the Euclidean space is bounded from above by an integral of its mean curvature with equality if and only if the domain is a ball. Combining with the isoperimetric inequality, one concludes that the volume of the domain is bounded by the integral of the mean curvature as well. Here, we will discuss a generalization of the latter inequality to complete manifolds with Ricci curvature bounded from below. As an application, one concludes that a complete manifold with sufficiently large bottom spectrum relative to its Ricci curvature lower bound admits no compact minimal hypersurfaces. In particular, this is the case for conformally compact Einstein manifolds whose boundaries at infinity have nonnegative scalar curvature.

In this talk, we present monotone quantities along the level set flow of $p$-capacitary functions in asymptotically flat $3$-manifolds with nonnegative scalar curvature. As applications, we prove geometric inequalities associated with $p$-capacitary functions, with rigidity on spatial Schwarzschild manifolds outside rotationally symmetric spheres, which generalizes Miao’s result from $p=2$ to $1<p<3$. Moreover, we recover mass-to-$p$-capacity and $p$-capacity-to-area inequalities due to Bray-Miao and Xiao. Compare to $p=2$ case, there is no conformal relationship for general $p$-capacitary functions between Euclidean and Schwarzschild model. The monotonicity property follows from a direct analysis of a system of ODEs arising from $p$-capacitary functions in Schwarzschild model. This is joint work with Jiabin Yin and Xingjian Zhou.

Ling Xiao (University of Connecticut)

Non-convexity of level sets for k-Hessian equations in convex ring

Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. In this talk, we will discuss the progress on the so-called Type IIA flows, introduced in a joint work with Fei, Phong and Picard, aiming to study the Type IIA equations from the flux compactifications of superstrings.