Informal Geometric Analysis Seminar

We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold. We prove a priori estimates for these metrics conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the K\"ahler setting. In addition, we show how these estimates can be used to prove a convergence result for a Hermitian analogue of the Calabi flow on compact complex manifolds with vanishing first Bott-Chern class. 

In this talk, I will show how Price inequalities for harmonic forms combined with some standard topology and geometry of 3-manifolds imply the Singer conjecture in dimension three. This provides an alternative proof of a result of Lott and Lueck (Invent. Math., 1995). Finally, I will outline some generalizations of this new approach in higher dimensions. This is part of a joint project with M. Hull and M. Stern.

The self-dual U(1)-Yang-Mills-Higgs functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly in the 2-dimensional and Kaehler settings) are objects of long-standing interest in low-dimensional gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.

In this talk, I will present some recent work on the uniqueness of shrinking gradient K\"ahler-Ricci solitons on non-compact toric manifolds. In particular, the familiar Delzant classification holds in this context, and this allows one to apply the techniques of Berman-Berndtsson, Wang-Zhu, and others which reduce the problem to a study of a particular real Monge-Amp\`{e}re equation on a convex domain in \mathbb{R}^n. As a consequence, we will see that the standard product of the Fubini-Study metric on \mathbb{CP}^1 (round metric on S^2) and Euclidean metric on \mathbb{C} is the only shrinking gradient K\"ahler-Ricci soliton on \mathbb{CP}^1 \times \mathbb{C} with bounded scalar curvature.

After introducing some background about stable bundles and HYM connections, I will explain both the analytic and algebraic sides when studying singularities of HYM connections. It turns out that local algebraic invariants can be extracted to characterize the analytic side. In particular, the analytic tangent cone is an algebraic invariant. (Based on joint works with Song Sun.)

A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation det(D^2u) = 1 are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.

In this talk, I will talk about Hardy Spaces for variety-deleted domains. Hardy Spaces have been studied extensively in different settings. I will present one of the main uses of Hardy Spaces on Several Complex Variables: to introduce boundary integral representation formulas for holomorphic functions. I will motivate our definitions and results with examples throughout the talk. Joint work with A.-K. Gallagher, P. Gupta and L. Lanzani.

Let M be a compact 3-manifold with scalar curvature at least 1. We show that there exists a Morse function f on M, such that every connected component of every fiber of f has genus, area and diameter bounded by a universal constant. The proof uses Min-Max theory and Mean Curvature Flow. This is a joint work with Davi Maximo. Time permitting, I will discuss a related problem for macroscopic scalar curvature in metric spaces (joint with Boris Lishak, Alexander Nabutovsky and Regina Rotman).

Generalized Kahler (GK) structures were first introduced in physics in the attempt to incorporate torsion into supersymmetric sigma models, and were independently discovered later in the works of Hitchin and Gualtieri. In this talk we explicitly describe invariant GK structures on toric Fano manifolds, and analyze the behaviour of the Generalized Kahler-Ricci Flow (GKRF) on these backgrounds. We prove that similarly to the Kahler case, GKRF admits a scalar reduction to a parabolic Monge-Ampere-type equation, allowing us to establish the long time existence of the normalized flow. Using the modification of the Perelman's W-functional, we prove that the only possible smooth limits are the usual Kahler-Ricci solitons. We suggest a generalization of Mabuchi energy to the GK setting, and use it to prove a week conditional convergence of the normalized GKRF on the level of potentials.