Johns Hopkins Geometric Analysis Seminar

Title: Circle bundles with PSC over large manifolds

Abstract: In his seminal work on metric inequalities for scalar curvature, Gromov asked whether total spaces of circles bundles over enlargeable manifolds can admit metrics with positive scalar curvature. We answer this question in all dimensions and construct infinitely many such examples over manifolds dimension 4 and above, and also that this is not possible over manifolds of dimension 3 and below. These examples also exhibit an unusual drop in macroscopic dimension. Our constructions are based on Donaldson's almost complex submanifolds. This is joint work with B. Sen.

Title : Scalar curvature comparison and rigidity of 3-dimensional weakly convex domains

Abstract: I will discuss a comparison and rigidity result of scalar curvature and scaled mean curvature on the boundary for weakly convex domain in Euclidean space, which is a joint work with Xuan Yao. This result is a smooth analog of Gromov's dihedral rigidity conjecture. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.

Title: Non-uniqueness of mean curvature flow

Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. With Xinrui Zhao, we show that even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Our examples extend to higher dimensions, complementing the surface examples obtained by Ilmanen and White. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish a generalized avoidance principle. We prove that level set flows satisfy this principle in the absence of non-uniqueness.

Abstract: Lecture 1 (lecture 2 will take place on March 24th at the Analysis & PDE seminar, see https://sites.google.com/view/hopkins-pde-seminar for details)

In this series of lectures, we will explore viscosity solutions to fully nonlinear elliptic equations, following the foundational book of L. Caffarelli and X. Cabr´e in Fully Nonlinear Elliptic Equations (AMS Colloquium Publications, 1995). Our primary goal is to introduce key concepts in the regularity theory of viscosity solutions, focusing on fundamental estimates and techniques. The lectures will be accessible to graduate students with a background in partial differential equations (PDEs) and analysis (basic measure theory and L p spaces). The material will be largely self-contained, making it suitable for those new to the subject. We will begin by discussing extremal classes of solutions (S), covering fundamental properties such as the Alexandroff-Bakelman-Pucci (ABP) estimate, Harnack inequality, and H¨older regularity. We will then study the regularity of solutions to the homogeneous equation with fixed coefficients, establishing uniqueness and C 1,α regularity. Finally, we will extend these results to the inhomogeneous equation with variable coefficients, obtaining interior C 1,α estimates. A tentative outline of the lectures is as follows: 

• Introduction: Definition of viscosity solutions, uniform ellipticity, Pucci’s extremal operators and the S class of solutions;
• Properties of the S class: the ABP estimate, weak L ε estimates, Harnack inequality, and C α regularity (Krylov-Safonov theory).
• Study of the homogeneous equation with constant coefficients: Uniqueness via Jensen’s method and C 1,α regularity.
• Study of the inhomogeneous equation with variable coefficients: Interior C 1,α estimates.

Title: Finite-Time Singularities of the Ricci Flow on Kähler Surfaces

Abstract: By work of Song-Weinkove, it is understood that the Ricci flow on any Kähler surface can canonically be continued through singularities in a continuous way until its volume collapses. This talk will discuss recent progress in understanding a more detailed picture of the singularity formation in this context.

Abstract: Lecture 3 (lecture 2 will take place on March 24th at the Analysis & PDE seminar, see https://sites.google.com/view/hopkins-pde-seminar for details)

In this series of lectures, we will explore viscosity solutions to fully nonlinear elliptic equations, following the foundational book of L. Caffarelli and X. Cabr´e in Fully Nonlinear Elliptic Equations (AMS Colloquium Publications, 1995). Our primary goal is to introduce key concepts in the regularity theory of viscosity solutions, focusing on fundamental estimates and techniques. The lectures will be accessible to graduate students with a background in partial differential equations (PDEs) and analysis (basic measure theory and L p spaces). The material will be largely self-contained, making it suitable for those new to the subject. We will begin by discussing extremal classes of solutions (S), covering fundamental properties such as the Alexandroff-Bakelman-Pucci (ABP) estimate, Harnack inequality, and H¨older regularity. We will then study the regularity of solutions to the homogeneous equation with fixed coefficients, establishing uniqueness and C 1,α regularity. Finally, we will extend these results to the inhomogeneous equation with variable coefficients, obtaining interior C 1,α estimates. A tentative outline of the lectures is as follows: 

• Introduction: Definition of viscosity solutions, uniform ellipticity, Pucci’s extremal operators and the S class of solutions;
• Properties of the S class: the ABP estimate, weak L ε estimates, Harnack inequality, and C α regularity (Krylov-Safonov theory).
• Study of the homogeneous equation with constant coefficients: Uniqueness via Jensen’s method and C 1,α regularity.
• Study of the inhomogeneous equation with variable coefficients: Interior C 1,α estimates.

Title: Stability Theorems for the Width 

Abstract: In this talk, I'll discuss some recent and ongoing work about the stability of min-max widths of spheres under various lower curvature bounds. Some of this is joint with Davi Maximo and Paul Sweeney Jr.

Title: Variations of the Yang-Mills Lagrangian in high dimension

Abstract: In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2-control of their curvature. We will start from the classical theory in critical dimension (i.e. n=4) and then move to the super-critical dimension (i.e. n>4), making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension.

Title: Rigidity of ancient ovals in mean curvature flow.

Abstract: We will discuss the classification of ancient noncollapsed flows as singularity models of mean curvature flow. In particular, I will discuss my recent joint work with Beomjun Choi and Jingze Zhu about spectral rigidity, stability and symmetry of the so-called k-ovals in general dimensions. These k-ovals belong to the family of ancient ovals (compact non-selfsimilar ancient noncollapsed flows), and are expected to coincide with all the ancient ovals by a recent work of Choi-Haslhofer.