The seminar usually takes place in 515, Cosmology Building at 4-5PM every Tuesday, 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)
Tang-Kai Lee 李堂愷 (Columbia University): Uniqueness problems in mean curvature flow
We will define what it means for a mean curvature flow evolution to be unique. We will discuss specific geometric conditions that guarantee a unique flow, as well as scenarios like "fattening" where uniqueness fails and the flow becomes non-deterministic.
Tathagata Ghosh (National Center for Theoretical Sciences): Moduli Spaces of Instantons on Asymptotically Conical Spin(7)-Manifolds
In this talk we discuss instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. After discussing the preliminary notions of G2 & Spin(7)-manifolds, asymptotically conical manifolds, and Yang-Mills equations & instantons, we mainly focus on the deformation theory of AC Spin(7)-instantons by relating the deformation complex with Dirac operators and spinors, and applying spinorial methods to identify the space of infinitesimal deformations with the kernel of a twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold.
As examples, we consider two important Spin(7) manifolds: $\mathbb{R}^8$, where $\mathbb{R}^8$ is considered to be an asymptotically conical manifold asymptotic to the cone over the round 7-sphere, and Bryant-Salamon manifold - the negative spinor bundle over 4-sphere, asymptotic to the cone over the squashed 7-sphere. We apply the deformation theory to describe deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on $\mathbb{R}^8$, and the Clarke-Oliviera instanton on the negative spinor bundle over 4-sphere. We also calculate the virtual dimensions of the moduli spaces using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operators.
If time permits, I will give an overview of my current work in understanding the moduli spaces better, which is based on joint work with D. Harland.
Albert Wood (Chinese University of Hong Kong): Symmetric Lagrangian Translators in C^n
Lagrangian mean curvature flow is the name given to the fact that mean curvature flow preserves the class of Lagrangian submanifolds in a Calabi-Yau manifold. Translating solutions to Lagrangian mean curvature flow play an important role in modeling singularities of the flow.
In this talk, I will introduce the topic of Lagrangian translators, and present a new family of explicit Lagrangian translators in C^m, constructed via a cohomogeneity-two ansatz that generalizes earlier work of Castro and Lerma. The solitons we obtain are invariant under the action of compact subgroups of SU(m) and arise from solutions to a reduced ODE system in C^2. We also discuss new symmetric examples of special Lagrangians, obtained by the same methods.
Kwok-Kun Kwong 鄺國權 (University of Wollongong): Fenchel-Willmore inequality for submanifolds in manifolds with non-negative k-Ricci curvature
In this talk, I will prove a sharp Fenchel-Willmore inequality for closed immersed submanifolds of arbitrary dimension and codimension in a complete Riemannian manifold with non-negative intermediate Ricci curvature. In the hypersurface case, this condition reduces to non-negative Ricci curvature. The result extends relatively recent work of Agostiniani, Fogagnolo, and Mazzieri, as well as classical results of Chen, Fenchel, Willmore, and others. If time permits, I will also present an unexpected connection between this inequality and a logarithmic Sobolev inequality on submanifolds. This is joint work with Meng Ji.
Boyu Zhang 章博宇 (University of Maryland at College Park): On the topology of stable minimal surfaces in PSC 4-manifolds
It is known that a closed 2-sided stable minimal hypersurface in a 4-manifold with positive scalar curvature (PSC) must be Yamabe positive, and hence it is diffeomorphic to a connected sum of S^1 x S^2’s and spherical space forms. We show that using a new compactness result for minimal surfaces in covering spaces and techniques from 4-manifold topology, one can obtain further control on the topology of stable minimal hypersurfaces. As an application, we show that the outermost apparent horizons of a smooth, asymptotically flat manifold with nonnegative scalar curvature must be diffeomorphic to S^3 or connected sums of S^1 x S^2’s. This is an extension of Hawking’s black hole topology theorem to dimension 4. The talk is based on joint work with Chao Li.
Thibault Langlais (Humboldt-Universität in Berlin): Geometry and incompleteness of G2-moduli spaces
G2-manifolds are an exceptional class of Ricci-flat manifolds occurring in dimension 7. Of particular interest are compact G2-manifolds, which play a similar role in M-theory in physics as Calabi-Yau manifolds do in string theory. It has been known for a long time that compact G2-manifolds are not rigid: they form finite-dimensional families of deformations, which can be globally parametrised by a moduli space. An important feature of these moduli spaces is that they come equipped with a natural metric, analogous to the Weil—Petersson metric, whose properties remain poorly understood. The goal of this talk will be to give a brief review of these topics and present new results concerning the incompleteness and the local geometry of the moduli spaces.
Xiaoxiang Chai 柴小祥 (POSTECH): Spectral curvature rigidity for bands
The Bonnet–Myers theorem asserts that a complete manifold is compact when the Ricci curvature is uniformly positive. In the case of a torical band, the product of a torus with an interval, Gromov showed that the width is bounded if the scalar curvature is positive. In this talk, I report band width estimates in the settings of spectral curvature, defined as the lowest eigenvalue of an elliptic operator involving the Laplace–Beltrami operator and the curvature terms. This is joint work with Yukai Sun (PKU).
Lan-Hsuan Huang 黃籃萱 (University of Connecticut): Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers
We present recent results on ADM mass–minimizing initial data sets: first, the equality case of the positive mass theorem, and second, Bartnik’s stationary vacuum conjecture arising from his quasi-local mass program.
A key new ingredient is a monotonicity formula for the Lorentzian length of a causal Killing vector field. This is joint work with Sven Hirsch.
Dongyeong Ko (Massachusetts Institute of Technology): Construction of minimal hypersurfaces with large first Betti number
We prove the regularity of cohomogeneity two equivariant isotopy minimization problems. We develop the cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in the 1980s. As an application of the theory, for $g \ge 1$ and $4 \le n+1 \le 7$, we construct minimal hypersurfaces $\Sigma_{g}^{n}$ on round spheres $\mathbb{S}^{n+1}$ with $(SO(n-1) \times \mathbb{D}_{g+1})$-symmetry. For sufficiently large $g$, $\Sigma_{g}^{n}$ is a sequence of minimal hypersurfaces with arbitrary large Betti numbers with topological type $\sharp^{2g} (S^{1} \times S^{n-1})$ or $\sharp^{2g+2} (S^{1} \times S^{n-1})$, which converges to a union of $\mathbb{S}^{n}$ and a Clifford hypersurface $\sqrt{\frac{1}{n}}\mathbb{S}^{1} \times \sqrt{\frac{n-1}{n}} \mathbb{S}^{n-1}$ or $\sqrt{\frac{2}{n}}\mathbb{S}^{2} \times \sqrt{\frac{n-2}{n}} \mathbb{S}^{n-2}$. In particular, for dimensions $5$ and $6$, $\Sigma_{g}^{n}$ has a topological type $\sharp^{2g} (S^{1} \times S^{n-1})$.
Taehun Lee 이태훈 (Konkuk University): Ancient mean curvature flows with finite total curvature
Ancient flows have been intensively studied over the past decade as singularity models for the mean curvature flow. In the spirit of the parabolic Liouville-type theorem for the non-compact case, flows with prescribed asymptotic behavior play an important role. In this context, we present a family of ancient mean curvature flows that converge to a given two-sided complete embedded minimal hypersurface in Euclidean space. We establish that these flows possess geometric properties such as finite total curvature, finite mass drop, and mean convexity for one family of these flows. This work is joint with Kyeongsu Choi (KIAS) and Jiuzhou Huang (KIAS).
Ao Sun 孙奥 (Lehigh University): Geometry of Mean Curvature Flow near Cylindrical Singularities
The cylindrical singularities are prevalent but complicated in geometric flows. We discuss one of the simplest extrinsic flow, the mean curvature flow, and illustrate how the local dynamics of the singularities influence the singular set itself, and the geometry and topology of the flow. This talk is based on joint works with Zhihan Wang (Cornell) and Jinxin Xue (Tsinghua).
Yohei Sakurai 櫻井陽平 (Saitama University): Almost splitting and quantitative stratification for super Ricci flow
I will discuss almost rigidity properties of super Ricci flow satisfying the so-called D-condition introduced by Buzano. I will present almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. This talk is based on the joint work with Keita Kunikawa (Tokushima university).
Kuan-Hui Lee 李冠輝 (McGill University) The Classification of non-Kähler Calabi Yau Geometries on threefolds
Calabi-Yau manifolds are fundamental geometry objects in complex geometry and they have many important results. Currently, these profound results have been extended to more general settings in complex geometry. In this talk, I will first introduce Bismut--Hermitian--Einstein manifolds (BHE) which can be viewed as a natural generalization of Calabi Yau manifolds in the non-Kähler setting. Next, I will provide some examples of BHE manifolds and show that a non-Kähler compact BHE surface must be a Hopf surface. In general, the BHE manifolds could be locally viewed as a R^2-principal bundle. The main purpose is to study the transverse geometry and prove a classification result under some topological constraint.
