The seminar usually takes place in 515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Nicolau Sarquis Aiex 艾尼克 (NTNU), Siao-Hao Guo 郭孝豪 (NTU), Chung-Jun Tsai 蔡忠潤 (NTU)
Sean McCurdy (National Taiwan Normal University): Reifenberg-type regularity for varifolds with critical mean curvature
One of the most famous results for regularity of varifolds is Allard's Regularity Theorem, which says that if a varifold $V$ is ``very close" to a single $m$-disc in $B_1^n(0)$ and has generalized mean curvature $\textbf{h} \in L^p(||V||)$ for $p>m$ then $V$ is a $C^{1, 1-p/m}$ graph in a neighborhood of the origin. This talk will focus upon what analogs of Allard's result hold in the \textit{critical case} $p=m$. After surveying previous results in local or infinitesimal regularity and in $m=p=2$ dimensions, we will present recent results in arbitrary dimension. If time permits, I will briefly mention some interesting ways in which these results point to a lacuna in the theory of functions on varifolds.
Simon-Raphael Fischer (National Center for Theoretical Sciences): Classification of neighbourhoods around leaves of a singular foliation
This talk is about my recent work with Camille Laurent-Gengoux. I will present our results about classifying singular foliations admitting a given leaf $L$ in a manifold $M$ and a given transverse model $(\mathbb{R}^d, \tau)$, where $\mathbb{R}^d$ is the fibre of a normal bundle of $L$ in $M$, and $\tau$ is a singular foliation in $\mathbb{R}^d$ admitting 0 as a leaf. Such a classification is motivated by the fact that every foliation $\mathcal{F}$ induces a singular foliation in the fibres of a normal bundle, the \emph{transverse (singular) foliation}, and these transverse foliations at each point in $L$ are canonically isomorphic. These isomorphisms are given by the parallel transport of what one calls $\mathcal{F}$-connections. The idea of this talk is to recover $\mathcal{F}$ given $(\mathbb{R}^d, \tau)$, and we will see that in a local neighbourhood around $L$ every foliation admitting $(\mathbb{R}^d, \tau)$ as transverse model is given by a certain infinite-dimensional principal bundle.
Jingwen Chen 陈景文 (University of Pennsylvania): Mean curvature flow with multiplicity 2 convergence
Mean curvature flow (MCF) has been widely studied in recent decades, and higher multiplicity convergence is an important topic in the study of MCF. In this talk, we present two examples of immortal MCF in $\mathbb{R}^3$ and $S^n \times [-1,1]$, which converge to a plane and a sphere $S^n$ with multiplicity 2, respectively. Additionally, we will compare our example with some recent developments on the multiplicity one conjecture and the min-max theory. This is based on joint work with Ao Sun.
Yang Li (Massachusetts Institute of Technology): On the Donaldson-Scaduto conjecture
Part of the Donaldson-Scaduto conjecture is concerned with constructing certain special Lagrangians inside the Calabi-Yau 3-folds obtained from products of C with an An type hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical special Lagrangians. This is joint work with Saman Esfahani.
Myles Workman (University College London): Minimal Hypersurfaces: Bubble Convergence and Index
The regularity theories of Schoen--Simon--Yau and Schoen--Simon for stable minimal hypersurfaces are foundational in geometric analysis. Using this regularity theory, in low dimensions, Chodosh--Ketover--Maximo and Buzano--Sharp studied singularity formation in sequences of minimal hypersurfaces through a bubble analysis.
I will review this background, before talking about my recent work in this bubble analysis theory. In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble converging sequence of minimal hypersurfaces.
Hsuan-Yi Liao 廖軒毅 (National Tsing Hua University): Atiyah classes of Lie pairs and of dg Lie algebroids
In this talk, I will start with a short introduction to Lie algebroids and will discuss Atiyah classes in two directions --- Lie (algebroid) pairs and dg Lie algebroids. The main theorem is that the Atiyah class of a Lie pair can be identified with the Atiyah class of the associated pullback dg Lie algebroid.
Yohsuke Imagi 今城洋亮 (ShanghaiTech): On compact Calabi--Yau conifolds
Compact Calabi–Yaus conifolds mean compact Calabi–Yau varieties with isolated singularities equipped with Ricci-flat Kahler metrics decaying to cone metric at the singularities. There was a (rather old) project by Yat-Ming Chan who carried out (by the gluing method) some simultaneous smoothings and resolutions of Calabi–Yaus and special Lagrangians. This was rather too technical at that time because no examples of compact Calabi–Yaus conifolds were known. Examples have now been found by Hans-Joachim Hein and Sun Song (using the result of Donaldson--Song). I will try to give a more systematic version of Chan's project.
Wu-Hsiung Huang 黃武雄 (National Taiwan University): Singularities and topological change for deforming domains in manifolds
Consider domains $D(t)$ deforming along $t$ on a hypersurface $M^n$ of constant mean curvature in $\mathbb{R}^{n+1}$.
If the topology of $D(t)$ is allowed to change along $t$ in some way, can the behavior of analysis maintain continuous in $t$, so that techniques in analysis still work?
For example, are the eigenvalues of the stability operator $L$ continuous in $t$? Sobolev continuity is also treated, as another related analysis entity.
Applications of this problem provides a global Morse index theorem, as well as the distribution of Jacobi fields along $t$-axis.
The results are carried over to a more popular case that the deformation is for the domains in general manifolds with a strongly elliptic adjoint operator $L$. It modifies Smale’s index theorem to a large extent that we can construct a route of deformation from a small ball to any pre-assigned domain.
The difficulty of the problem is focused on the singularities appeared when the topology is changed.
A simple example is that a small geodesic disk on a cylinder $M^2$ enlarges with its increasing radius until the two ends on the boundary circle touches at a singularity $q$, which is the joint point of the two ends. If the domain continues to enlarge, it becomes a ring domain. The topology of a disk now changes to a ring. Are the eigenvalues still continuous as the domain crosses the singularity $q$?
Ryosuke Takahashi 楊劼之 (National Cheng Kung University): Generalization of Atiyah-Patodi-Singer Index Theorem
In 1975, M. Atiyah, V. Patodi and I. Singer generalized Atiyah-Singer index theorem on manifolds with boundary. It describes the index for an elliptic boundary problem by the integral of $\hat{A}$-genus on $M$ and an eta invariant defined on $\partial M$. A Further generalization was studied by Weichung Chou, Fangyun Yang, which describes the index theorem on manifolds with degenerate boundaries. In this talk, we will discuss a more general case: Manifolds with stratified boundaries and the corresponding index theorems.
Zhou Jie 周杰 (Capital Normal University) : Optimal rigidity estimates of varifolds almost minimizing the Willmore energy
In this presentation, we talk about the stability of the Willmore functional. For an integral 2-varifold $V=\underline{v}(\Sigma,\theta)$ in $R^n$ with square integrable generalized mean curvature andfinite mass. If its Willmore energy is smaller than $4\pi(1+\delta^2)$ and the mass is normalized to be $4\pi$, we show that $\Sigma$ is $W^{2,2}$ and bi-Lipschitz close to the round sphere in a quantitative way when $\delta<\delta_0\ll1$. For $n=3$, we show the sharp constant is $\delta_0^2=2\pi$. This is a joint work with Dr. Yuchen Bi.
Pei-Ken Hung 洪培根 (University of Illinois Urbana-Champaign): Toroidal positive mass theorem
I will discuss the positive mass theorem for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity. The rigidity is obtained provided the second fundamental form is umbilic. Without the umbilic assumption, the rigidity fails and we construct a counterexample based on the Horowitz-Myers geon. We also recover the local rigidity theorem of Eichmair-Galloway-Mendes in dimension 3. These results are obtained through an analysis of spacetime harmonic functions. This is a joint work with Aghil Alaee and Marcus Khuri.
Brian Harvie (NCTS): Weak Inverse Mean Curvature Flow and Minkowski Inequalities in Hyperbolic Space
Inverse mean curvature flow (IMCF) is a geometric flow that expands hypersurfaces by mean curvature. The flow has many geometric applications, especially to geometric inequalities, but a key obstacle to these is the formation of finite-time singularities. To deal with this, Huisken and Ilmanen developed a notion of weak solutions of IMCF which flow through these singularities and out to infinity. Weak IMCF is well-understood in Euclidean space, but much less is known about weak IMCF in hyperbolic space.
In this talk, I will present my recent work about regularity of weak IMCF in hyperbolic space, the key to which is an Alexandrov reflection method in the Poincare ball. Then I will apply these regularity results to extend two Minkowski inequalities for hypersurfaces in hyperbolic space. One of these inequalities implies a certain Penrose-type inequality in general relativity.
