Differential and Complex Geometry Seminar

Organized by Mario Chan, Eunjoo Lee, Hojoo Lee, and Hyung-Seok Shin

[24] July 31 (Friday) 2015
• Speaker: Farid Madani (Goethe University Frankfurt)
• Title: S^1-Yamabe invariant on 3-manifolds
• Abstract: After a short overview on the (non-equivariant) Yamabe invariant, we introduce the equivariant one. We show that the S^1-Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. Moreover, we give a topological upper bound for the S^1-Yamabe invariant of any closed oriented 3-manifold endowed with a circle action. This is joint work with Bernd Ammann and Mihaela Pilca.

• [23] July 31 (Friday) 2015
• Speaker: Soojung Kim (Institute of Mathematics, Academia Sinica, Taiwan)
• Title: Regularity for elliptic equations on Riemannian manifolds II
• Abstract: The Krylov-Safonov type Harnack inequality for elliptic operators on Riemannian manifolds was initiated by Cabre [1], based on the ABP type estimates. In these lectures, I will explain the Krylov-Safonov regularity estimates for viscosity solutions to degenerate and singular operators of p-Laplacian type on Riemannian manifolds with Ricci curvature bounded from below.

References:

(1) X. Cabre, Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm. Pure. Math. 50 (1997), 623-665.

(20 L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.

(3) S. KIm, Second derivative estimates for uniformly elliptic operators on Riemannian manifolds, Nonlinear Anal. TMA 112 (2015), 215-231.

(4) S. Kim, Harnack inequality for degenerate and singular operators of p-Laplacian type on Riemannian manifolds, arXiv:1503.09032

• [22] July 30 (Thursday) 2015
• Speaker: Soojung Kim (Institute of Mathematics, Academia Sinica, Taiwan)
• Title: Regularity for elliptic equations on Riemannian manifolds I
• Abstract: The Krylov-Safonov type Harnack inequality for elliptic operators on Riemannian manifolds was initiated by Cabre [1], based on the ABP type estimates. In these lectures, I will explain the Krylov-Safonov regularity estimates for viscosity solutions to degenerate and singular operators of p-Laplacian type on Riemannian manifolds with Ricci curvature bounded from below.

References:

(1) X. Cabre, Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm. Pure. Math. 50 (1997), 623-665.

(2) L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.

(3) S. KIm, Second derivative estimates for uniformly elliptic operators on Riemannian manifolds, Nonlinear Anal. TMA 112 (2015), 215-231.

(4) S. Kim, Harnack inequality for degenerate and singular operators of p-Laplacian type on Riemannian manifolds, arXiv:1503.09032

• [21] June 18 (Thursday) 2015
• Speaker: Yuuji Tanaka (National Center of Theoretical Sciences, National Cheng Kung University, Taiwan)
• Title: A construction of Spin(7)-instantons
• Abstract: In this talk, we describe a construction of Spin(7)-instantons on examples of compact Spin(7)-manifolds by Joyce. Spin(7)-instantons are elliptic Yang-Mills connections on eight-dimensional manifolds with holonomy contained in the group Spin(7), which minimize the Yang-Mills energy. Analytic properties of Spin(7)-instantons have been studied by Gang Tian and others, but little was known about the existence of examples of Spin(7)-instantons on compact Spin(7)-manifolds other than an Oxford Ph.D thesis by Christopher Lewis in 1998. There are two known constructions of compact Spin(7)-manifolds both obtained by Dominic Joyce. The first one begins with a torus orbifold of a special kind with non-isolated singularities. The Spin(7)-manifold is obtained by resolving the singularities with the aid of algebraic geometry techniques. The second one begins with a Calabi-Yau four-orbifold with isolated singular points of a special kind and an anti-holomorphic involution fixing only the singular points. The Spin(7)-manifold is obtained by gluing ALE Spin(7)-manifolds with anti-holomorphic involutions fixing only the origins to each singular point. Lewis studied the problem of constructing Spin(7)-instantons on Spin(7)-manifolds coming from Joyce's first construction. More recently, Thomas Walpuski proved a rather general construction of Spin(7)-instantons as well as constructions of G2-instantons in his Ph.D thesis, which includes the example obtained by Lewis. This talk describes a construction of Spin(7)-instantons on compact Spin(7)-manifolds coming from Joyce's second construction. The Spin(7)-manifold is obtained by gluing ALE Spin(7)-manifolds at each singular point of a Calabi-Yau four-orbifold with finitely many singular points, and an anti-holomorphic involution fixing only the singular points. Assuming that there are Hermitian-Einstein connections with certain conditions on both the Calabi-Yau four-orbifold and the ALE Spin(7)-manifolds, we glue them together simultaneously with the underlying pieces to obtain a Spin(7)-instanton on the compact Spin(7)-manifold.

• [20] May 22 (Friday) 2015
• Speaker:  Leobardo Rosales (Keimyung University)
• Title: Adventures in Geometric Measure Theory II
• Abstract: A journey to Allard's Regularity Theorem.

The subject of Geometric Measure Theory was invented in the process of answering, in the affirmative, the problem of Plateau: can we fit a surface of minimal area spanning a given curve in space? One tool in GMT developed as a consequence of considering Plateau's problem is the concept of varifolds, which we can define as distributions which almost everywhere correspond to submanifolds. In particular, we consider stationary varifolds, which are the versions of minimal submanifolds for varifolds. The natural question to consider is then to give criteria sufficient to conclude when a stationary varifold corresponds to a classical minimal submanifold. One such criteria is Allard's Regularity Theorem, which states that wherever a stationary varifold has area sufficiently close to a plane, then the stationary varifold is a classical minimal submanifold. Our goal in this lecture series is to begin with a (special) definition of stationary varifolds, and then sketch a proof of Allard's Regularity Theorem.

• [19] May 20 (Wednesday) 2015
• Speaker: Leobardo Rosales (Keimyung University)
• Title: Adventures in Geometric Measure Theory I
• Abstract: A journey to Allard's Regularity Theorem.

The subject of Geometric Measure Theory was invented in the process of answering, in the affirmative, the problem of Plateau: can we fit a surface of minimal area spanning a given curve in space? One tool in GMT developed as a consequence of considering Plateau's problem is the concept of varifolds, which we can define as distributions which almost everywhere correspond to submanifolds. In particular, we consider stationary varifolds, which are the versions of minimal submanifolds for varifolds. The natural question to consider is then to give criteria sufficient to conclude when a stationary varifold corresponds to a classical minimal submanifold. One such criteria is Allard's Regularity Theorem, which states that wherever a stationary varifold has area sufficiently close to a plane, then the stationary varifold is a classical minimal submanifold. Our goal in this lecture series is to begin with a (special) definition of stationary varifolds, and then sketch a proof of Allard's Regularity Theorem.

• [18] April 27 (Monday) 2015
• Speaker: Tran Vu Khanh (University of Wollongong)
• Title: Pseudoconvex domains of infinite type
• Abstract: Pseudoconvexity is one of the most central concepts in modern complex analysis. There are two kinds of pseudoconvexity: finite type and infinite type. A lot of works have been done about pseudoconvexity of finite type. We are interested in the infinite type case. In this talk we present some recent results on pseudoconvex domains of infinite type. The content of the talk consists of:

1) The influence of geometric type (finite and infinite type) on estimates and regularity of the d-bar Neumann problem;

2) Boundary regularity of the solution to the complex Monge-Ampere equation on pseudoconvex domains of infinite type.

• [17] April 9 (Thursday) 2015
• Speaker: Sui-Chung Ng (East China Normal University, Shanghai)
• Title: Proper holomorphic mappings among type-I bounded symmetric domains
• Abstract: A continuous map is said to be proper if the preimage of every compact set is compact. In Several Complex Variables, proper holomorphic maps are among the most important mappings having been investigated for domains. Starting from 1980s, there has been a lot of work for the complex unit balls, basing mainly on the techniques from Cauchy-Riemann geometry. On the other hand, results for bounded symmetric domains of higher rank are rather limited due to the singularities of their boundaries. We aim to present in this talk a new approach to the proper holomorphic maps among bounded symmetric domains of higher rank. It makes use of certain geodesic subdomains of bounded symmetric domains and by that we can translate the original problem to a problem between generalized balls, which is much easier to work with.

• [16] March 30 (Monday) 2015
• Speaker: Takato Uehara (Saga University)
• Title: Isolated periodic points for area-preserving surface mappings
• Abstract: One of the important problems is to investigate the growth rate of the number of periodic points for a surface mapping. However the existence of curves of periodic points makes it difficult to determine the growth rate. In my talk, I will focus on area-preserving birational mappings (possibly admitting periodic curves) on projective complex surfaces, and determine the growth rate of isolated periodic points for the mappings.

• [15] January 30 (Friday) 2015
• Speaker: Kwokwai Chan (The Chinese University of Hong Kong)
• Title: An introduction to SYZ mirror symmetry
• Abstract: The SYZ conjecture suggests that mirror symmetry can be explained as a kind of geometric Fourier transform. In this talk, I will go through the SYZ constructions by looking at a particular example, namely, the local P^2. I hope the audience can at least get a rough idea of how mirror symmetry lead to enumerative predictions.

• [14] November 24 (Monday) 2014   4:00 PM @ Room 1423
• Speaker: Pak Tung Ho (Sogang University)
• Title: Differentiable rigidity of hypersurface in space forms
• Abstract: In this talk, I will talk about the results of my paper "Differentiable rigidity of hypersurface in space forms" which is available here. I will explain the idea of the proof in my paper.

• [13] November 24 (Monday) 2014    2:00 PM @ Room 1423
• Speaker: Pak Tung Ho (Sogang University)
• Title: Yamabe flow
• Abstract: First I will explain the definition of the Yamabe flow, which was introduced to study the Yamabe problem. Then I will talk about the properties of the Yamabe flow, especially the results of the convergence of the Yamabe flow. If time permits, I will also talk about the prescribing scalar curvature flow, which is a generalization of the Yamabe flow.

• [12] November 10 (Monday) 2014   2:00 PM @ Room 1423
• Speaker: Chanyoung Sung (Korea National University of Education)
• Title: Liouville-type theorems and its applications to geometry
• Abstract: On a (non-compact) complete manifold with a mild lower bound of Ricci curvature, we prove some Liouville-type theorems for a real-valued function satisfying a 2nd order elliptic inequality. It has some geometric consequences. For example, such a manifold cannot be minimally immersed in a bounded subset of a Cartan-Hadamard manifold.

• [11] November 3 (Monday) 2014   2:00 PM @ Room 1424
• Speaker: Filippo Morabito (Korea Advanced Institute of Science and Technology)
• Title: Higher genus capillary surfaces in a unit ball of R^3
• Abstract:  I will present the construction of minimal surfaces of positive genus which are embedded in a unit ball of R^3 and meet the boundary of the ball at locally constant contact angle along three closed curves.

• [10] November 3 (Monday) 2014   11:00 AM  @ Room 1424
• Speaker: Filippo Morabito (Korea Advanced Institute of Science and Technology)
• Title: Costa-Hoffman-Meeks minimal surfaces
• Abstract:  I will provide an introduction to the CHM surfaces.

• [09] September 12 (Friday) 2014   11:00 AM @ Room 1424
• Speaker: Leobardo Rosales (Korea Institute for Advanced Study)
• Title: Energy minimizing unit vector fields
• Abstract: Given a surface-with-boundary in space, we study the extrinsic energy of smooth tangent unit-length vector fields. Fixing continuous tangent unit-length vectors fields on the boundary of a given surface, we ask if there is a unique smooth tangent unit-length vector field continuously achieving the boundary data and minimizing energy amongst all smooth tangent unit-length vector fields also continuously achieving the boundary data.

• [08] August 12 (Tuesday) 2014   3:00 PM @ Room 1423
• Speaker: Carla Cederbaum (University of Tübingen)
• Title: On the definition of mass and center of mass of isolated systems in Newtonian gravity and general relativity
• Abstract: Isolated gravitating systems such as stars, black holes or galaxies play an important role both in Newton's theory of gravity(NG) and in general relativity (GR). While the definition of mass and center of mass via the mass density is straightforward in NG, GR knows several promising approaches. The most important definitions of center of mass in GR go back to Beig and O Mur- chadha/Arnowitt, Deser and Misner (BM/ADM) as well as to Huisken and Yau (HY). Under certain assumptions on the asymptotic decay, the BM/ADM and the HY centers coincide (Huang, Metzger-Eichmair, Nerz). However, both notions subtly depend on the chosen asymptotic coordinates; in particular, we will present an explicit example in which both centers diverge and a corresponding example in NG (C-Nerz). Moreover, we will relate the Newtonian and the relativistic centers with the help of the Newtonian limit (in the case of static systems).

• [07] August 12 (Tuesday) 2014   2:00 PM @ Room 1423
• Speaker: Julie Clutterbuck (Monash University)
• Title: Translating solitons for mean curvature flow
• Abstract: Translating solitons to mean curvature flow are interesting in both their own right, and as models for singularities. It is believed that, like minimal surfaces, there ought to be many examples. However only a few explicit ones are known. I will talk about joint work with Felix Schulze and Oliver Schnuerer on the stability of translating paraboloids, and also describe some more recent work.

• [06] August 12 (Tuesday) 2014   1:00 PM @ Room 1423
• Speaker:  Frank Morgan (Williams College)
• Title: New isoperimetric theorems and open questions
• Abstract: After 2000 years, the isoperimetric problem of minimizing perimeter for given volume remains open in most settings, including Riemannian manifolds, perhaps with density, perhaps with obstacles. We'll discuss some recent advances by Cabré, Rosales, Milman, Chambers, and collaborators and some open questions.

• [05] July 25 (Friday) 2014   11:00 AM @ Room 1424
• Speaker: Hojoo Lee (Korea Institute for Advanced Study)
• Title: Lagrangian solitons to mean curvature flow
• Abstract: We survey recent results on Lagrangian solitons to mean curvature flow in complex Euclidean space.

• [04] June 13 (Friday) 2014   5:00 PM @ Room 1424
• Speaker: Joe S. Wang
• Title: Symmetries of CMC surfaces
• Abstract: We introduce two classes of symmetries of the differential equation for CMC surfaces and its associated hierarchy. First, a formal Killing field is determined by the method of enhanced prolongation. Second, a non-local spectral Killing field is solved by a dressing transformation and an Abelian extension. We remark on the various applications of these symmetries to the theory of CMC surfaces.

• [03] May 26 (Monday) 2014   4:00 PM @ Room 1424
• Speaker: Chun-Chi Lin (National Taiwan Normal University)
• Title: On the L^2-flows of elastic curves with hinged ends
• Abstract: In this talk, we would like to discuss two types of evolution equations for open elastic curves with hinged ends and fixed total length. The bending energy of curves, \int \kappa^2 ds, decreases during both types of the geometric flows. These geometric flows could be viewed as one-dimensional Willmore flows. We would focus on the issue of long-time existence of smooth solutions, which would be obtained by energy estimates via interpolation inequalities. Proper boundary conditions play a crucial role in these estimates.

• [02] April 11 (Friday) 2014   11:00 AM @ Room 1424
• Speaker: Filippo Morabito (Korea Advanced Institute of Science and Technology)
• Title: Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds
• Abstract: I will present some results of Farina, Mari, Valdinoci about elliptic PDE's on Riemannian manifolds whose Ricci curvature satisfies some conditions. In particular, they showed that assuming the existence of a stable solution, it is possible to classify the solution and the manifold.

• [01] March 14 (Friday) 2014  11:00 AM @ Room 1424
• Speaker: Leobardo Rosales (Korea Institute for Advanced Study)
• Title: Two-dimensional solutions to the c-Plateau problem
• Abstract: The c-Plateau problem for surfaces in space asks, given $c>0$ and $\gamma$ a closed curve in space, whether we can find $M_{c}$ a smooth orientable surface-with-boundary, with $\partial M_{c} = \sigma_{c}+\gamma$ where $\sigma_{c}$ is a finite union of closed curves disjoint from $\gamma,$ minimizing $c$-isoperimetric mass $\mathbf{M}^{c}(M) := \text{area}(M)+c \cdot \text{length}(\partial M)^{2}$ amongst all $M$ smooth orientable surfaces-with-boundary, with $\partial M = \sigma+\gamma$ where $\sigma$ is a finite union of closed curves disjoint from $\gamma$? In this talk we give several regularity results for solutions to the $c$-Plateau problem, formulated in the more general setting of integer two-rectifiable currents.