The seminar usually takes place in Lecture Hall 5F, Cosmology Building at 3-4PM and/or 4-5PM every other Thursday, unless noted otherwise.
Organizers: Siao-Hao Guo 郭孝豪 (NTU), Ulrich Menne 孟悟理 (NTNU), Wei-Bo Su 蘇瑋栢 (Academia Sinica), Chung-Jun Tsai 蔡忠潤 (NTU)
Nick Edelen (University of Notre Dame): Degeneration of 7-dimensional minimal hypersurfaces with bounded index
A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitely-many diffeomorphism types.
Kuan-Hui Lee 李冠輝 (University of California, Irvine): The stability of generalized Ricci solitons
The generalized Ricci flow is first studied by Streets and Tian. It can be viewed as the Ricci flow of connections with torsion which has many applications in non-Kähler geometry. In this talk, we will study the fixed points of the generalized Ricci flow which are called the steady generalized Ricci solitons. Similar to Ricci flow, generalized Ricci flow is also a gradient flow so we can further compute the variation formulas and see some linear stable examples. Our main result is to show that the concept of dynamical stability and linear stability are equivalent on the steady gradient generalized Ricci solitons.
Ryosuke Takahashi 高橋 良輔 (Kyushu University): $J$-equation on holomorphic vector bundles
In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry. In this talk, we introduce the $J$-equation on holomorphic vector bundles over compact K\"ahler manifolds, as an extension of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental properties as well as examples. In particular, we give an algebraic stability condition called the (asymptotic) $J$-stability in terms of subbundles on compact K\"ahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.
Kwok-Kun Kwong 鄺國權 (University of Wollongong): On an inverse curvature flow in two-dimensional space forms
I am going to discuss the evolution of compact convex curves in two-dimensional space forms under an inverse curvature type flow, first proposed by Brendle, Guan and Li. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. This flow was studied by Brendle-Guan-Li and by Hu-Li-Wei in higher dimensions (hypersurfaces of dimension $n\ge 2$). The curve case ($n=1$) has been left open in earlier work and indeed the analysis of the flow must be done differently in this case. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Gir\~ao-Pinheiro. This is joint work with Y. Wei, G. Wheeler and V. Wheeler.
Yi-Sheng Wang 王以晟 (Academia Sinica): Characteristic diagram of a handlebody-knot
Johannson’s characteristic submanifold theory provides an effective and convenient way to decompose a three-manifold, and has been used extensively in studying automorphisms of a three- manifold. In this talk, I will explain how the theory can be applied to classifying handlebody-knots, which are embedded handlebodies in the 3-sphere, and computing their symmetry groups. The talk will begin with a quick review of characteristic submanifold theory, which is followed by a short introduction to the Koda-Ozawa classification of essential annuli in handlebody-knot exteriors. Based on them, the notion of a characteristic diagram of a handlebody-knot will be introduced, and a classification result on irreducible, atoroidal handlebody-knots, via characteristic diagrams, will be presented. I will wrap up the talk with some applications to handlebody-knot symmetry.
Eungbeom Yeon 연응범 (Pusan National University): Existence and uniqueness results on free boundary minimal surfaces via Weierstrass representation formula
There are various results on the existence and uniqueness of capillary minimal surfaces. Capillary minimal surface $\Sigma$ in a domain $U$ is a minimal surface $\Sigma$ which meets $\partial U$ in a constant contact angle along $\partial U$. If the contact angle is a right angle then such a surface is called a free boundary minimal surface. A famous Fraser-Li conjecture asks if the critical catenoid would be the only embedded free boundary minimal annulus in a unit ball in $\mathbb{R}^3$. In this talk, I will describe some of the recent results on the existence and uniqueness of capillary minimal surfaces mostly exploiting a well known Weierstrass representation formula. The talk will include some results and observations regarding Fraser-Li conjecture based on a joint work with Jaehoon Lee.
Giada Franz (ETH Zurich): Equivariant min-max theory to construct free boundary minimal surfaces in the unit ball
A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e., the unit sphere). A very natural question is whether there are FBMS in the unit ball of any given topological type.
In this talk, we will present the construction of a family of FBMS with connected boundary and arbitrary genus, via an equivariant version of Almgren-Pitts min-max theory à la Simon-Smith. We will see how this method allows us to control the topology of the resulting surface and also obtain information on its index.
