Geometry
Organizers: Otis Chodosh & Greg Parker
Upcoming Events
The Vafa Witten equations on 4-manifolds are the variational equations of a functional that generalizes one of the Chern-Simons functionals for SL(2;C) connections on 3-manifolds (and it reduces to that on products of a 3-manifold with the circle). Being that the moduli space of solutions…
The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a…
Hyperbolic reflection groups appear in various fields of mathematics such as algebraic geometry, discrete subgroups of Lie groups, geometric group theory, geometric topology, and number theory. Cofinite and cocompact hyperbolic reflection groups have the following feature: their fundamental…
Past Events
The existence problem for minimal hypersurfaces (or more generally prescribed mean curvature hypersurfaces) in complete noncompact manifolds is fundamental, but little is known. Min-max provides a powerful framework for existence in closed manifolds, but relies critically on compactness. I…
I will explain a certain topological construction of positive scalar curvature metrics with uniformly Euclidean ($L^\infty$) point singularities. This provides counterexamples to a conjecture of Schoen. It also shows that there are metrics with uniformly Euclidean point singularities which…
Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs, especially in relation to physical models. Since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, it is well known that the optimal regularity of a Riemannian structure is governed by…
Entire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type large-scale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blow…
We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n \geq 7 we illustrate sheeting results around "flat…
Lagrangian mean curvature flow is a potentially powerful tool for tackling problems related to symplectic topology. Joyce made a serious of conjectures about the singularity formation, surgeries and long-time behaviour in Lagrangian mean curvature flow. I will describe recent…
We are going to discuss the following generalization of the classical boxing inequality:Let $M^n$ be a manifold in a finite- or infinite-dimensional Banach space $B$, and $m\leq n$ a positive number.Then there exists a pseudomanifold $W^{n+1}$ in $B$ such that $\partial W^{n+1}=M^n$, and the $m…
In this talk, I will explain why fractional (or nonlocal) minimal surfaces are ideal objects to which min-max methods can be applied on Riemannian manifolds. After a short introduction about these objects and how they approximate minimal surfaces, I will present a vision for the future on how to…
I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in S^3 and free boundary minimal surfaces in B^3 via constrained optimization problems for Laplace and Steklov eigenvalues on surfaces. Along the way, I'll highlight…
Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-…