The Monge-Ampere equation det(D^2 u) = 1 arises in prescribed
curvature problems and in optimal transport. An interesting feature of the
equation is that it admits singular solutions. We will discuss new examples
of convex functions on R^n that solve the Monge-Ampere equation away from
finitely many points, but contain polyhedral and Y-shaped singular
structures. Along the way we will discuss geometric and applied motivations
for constructing such examples, as well as their connection to a certain
Typical comparison results in Riemannian geometry, such as for
volume or for spectrum of the Laplacian, require Ricci curvature lower
bounds. In dimension three, we can prove several sharp comparison estimates
assuming only a scalar curvature bound. The talk will present these results,
their applications, and describe how dimension three is used in the proofs.
Joint work with Jiaping Wang.
In this talk, I will present parts of a recent paper written
jointly with Mario Garcia-Fernandez and Jeff Streets ("NonKahler Calabi-Yau
Geometry and Pluriclosed Flow" arxiv:2106.13716). In particular, I intend to
discuss long-time existence and convergence of solutions to pluriclosed flow
on manifolds satisfying a flatness condition. As the pluriclosed flow does
not admit a scalar reduction in general, I will introduce the language of
holomorphic Courant algebroids and a result of J.M. Bismut that will allow
us to derive apriori estimates for an equivalent coupled
I will introduce the harmonic level set method developed by Stern in 2019.
This technique has been used to prove the positive mass theorems in various
settings, for example, the Riemannian case, the spacetime case, the
hyperbolic case, and the positive mass theorem with charge. I will focus on
the positive mass theorem for asymptotically hyperbolic manifolds. We give a
lower bound for the mass in the asymptotically hyperbolic setting. In this
setting, we solve the spacetime harmonic equation and give an explicit
expansion for the solution. We also prove some rigidity results as
corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri.
The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. It can also be used to prove a stability result about Kahler structures. This is based on joint work with Phong, Picard and Zhang.
The deformed Hermitian-Yang-Mills equation, which will be
abbreviated as dHYM equation, was discovered around the same time in the
year 2000 by Mariño-Minasian-Moore-Strominger and Leung-Yau-Zaslow using
different points of view.
In this talk, first, I will skim through Leung-Yau-Zaslow’s approach in a
simple way. Then I will introduce the C-subsolution which is introduced by
Székelyhidi and Guan, I will go over some known results of the dHYM
equation, and I will bring up my previous results. Last, I will show some of
my recent works which will appear soon.
To characterize scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass).
In Riemannian manifold $(M^n, g)$, it is well-known that its minimizing hypersurface is smooth when $n\leq 7$, and singular when $n\geq 8$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.
Schoen-Yau proved the spacetime positive energy theorem by reducing
it to the time-symmetric (Riemannian) case using the Jang equation. To
acquire solutions to the Jang equation, they introduced a family of
regularized equations and took the limit of regularized solutions, whereas a
sequence of regularized solutions could blow up in some bounded regions
enclosed by apparent horizons. They analyzed the blowup behavior near but
outside of apparent horizons, but what happens inside remains unknown. In
this talk, we will discuss the blowup behavior inside apparent horizons
through two common geometric treatments: dilation and translation. We will
also talk about the relation between the limits of blowup regularized
solutions and constant expansion surfaces.