Learning Seminar
Basic Research Laboratory, Geometry of Submanifolds
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General information
Book: A Course in Minimal Surfaces (T. Colding & W. Minicozzi, AMS, 2011/01)
Time: 10:00 am to 12:00 pm every Monday (GMT+9)
Venue: Comprehensive Research Bldg 313
Spring Semester in 2023
(12.05.Mon)
The theorey of varifolds I
Speaker : Jihyeon Lee
Abstract
We will describe the basic theory of varifolds in Euclidean space which is a generalization of submanifolds. First, we identify submanifolds with certain Radon measures using the compactness theorem of Radon measures. Then we define the stationary of varifolds by computing the first variation of varifolds. Second, we give the monotonicity formula and the mean value inequality for stationary varifolds.
(10.24.Mon)
Almost stability
Speaker : Sangwoo Park
Abstract
I will introduce "almost stability". We have focused on the estimates for stable minimal surfaces previous our seminar. It can be extended to surfaces that are almost stable in some sense. Then we will discuss their ideas and applications of it.
(10.17.Mon)
Various compactness theorems
Speaker : Eungbeom Yeon
Abstract
We look into Choi-Schoen's compactness theorem on minimal surfaces of fixed genus in 3-manifold with positive Ricci curvature. Based on the theorem, various compactness results with or without area and index bounds will be surveyed.
(10.06.Thu)
Curvature estimate for stable minimal hypersurface and its application II
Speaker : Yuan Shyong
Abstract
We first discuss how having area bound on 2D minimal surface gives us some control on its curvature. For higher dimension, using Simons’ inequality and stability inequality, we can derive L^p curvature estimate for stable minimal hypersurface and use it to show Bernstein type result. If time permits, we shall also discuss a criterion for stablility of minimal hypersurface.
(09.26.Mon)
Curvature estimate for stable minimal hypersurface and its application I
Speaker : Yuan Shyong
Abstract
We first discuss how having area bound on 2D minimal surface gives us some control on its curvature. For higher dimension, using Simons’ inequality and stability inequality, we can derive L^p curvature estimate for stable minimal hypersurface and use it to show Bernstein type result. If time permits, we shall also discuss a criterion for stablility of minimal hypersurface.
(09.22.Thu)
On free boundary problem and minimal surfaces
Speaker : Yoonjung Lee
Abstract The free boundary problem, sometimes referred as an overdetermined problem, is to determine an unknown domain and an unknown solution to solve PDEs.
This paper shows a connection with a solution of the free boundary problem and minimal surface.
I will start with some motivation about this free boundary problem.
In this talk, the main goal is to prove Theorem 1 which shows a solvability of the free boundary problem by construct the solution based on the Simons' cone.
This result could provide a counter-example of the De Giorigi type conjecture (see Theorem 2 and 3) which is a PDE analogue of the Bernstein theorem.
This talk partially cover page 993-1009 in the following paper.
Youg Liu, Kelei Wang, Juncheng Wei
On a free boundary problem and minimal surface
Ann. I. H. Poincaré, AN 35, (2018), 993-1017.
(09.19.Mon)
Simons' inequality and small curvature estimates for mimimal surfaces
Speaker : Eungbeom Yeon
Abstract
We look into Simons' inequality for minimal hypersurfaces and discuss its geometric meaning.
We also start the big subject of curvature estimates with some of the classical results.
(09.15.Thu)
Analysis of minimal multi-valued graphs
Speaker : Sangwoo Park
Abstract
There are two local models for embedded minimal disks (by an embedded disk, we mean a smooth injective map from the closed unit ball in R^2 into R^3 ). One model is the plane (or, more generally, a minimal graph), the other is a piece of a helicoid. The latter case fails to be a graph, but is instead a "multi-valued graph".In this talk, I will focus on constructing local Examples of Multi-valued graphs
(09.05.Mon)
Bochner's formula and its applications in the geometry
Speaker : Eungmo Nam
Abstract
Bochner's formula is extremely useful formula in a Riemannian manifold. In this talk, I will give a proof of this formula based on the symmetry of hessian.
And then, I am going to give some generalizations and applications of this formula such as a weighted version of Bochner's formula and Reilly formula.
Read the appendix: Bochner's formula (p60~63)
In discussion session, I want to discuss a weighted version of Bochner's formula and Reilly formula.
(09.01.Thu)
The Gauss map of minimal surfaces in n-dimensional Euclidean spaces
Speaker : Eungbeom Yeon
Abstract
The Gauss map of minimal surfaces in n-dimensional Euclidean spaces have a lot of deep geometric properties. We look into a few of the properties and as the consequential results study theorem of Bernstein and Weierstrass representation formula for minimal surfaces in 3-dimensional Euclidean spaces
(08.22.Mon)
Minimal submanifolds, and the first and second variational formulae
Speaker : Jihyeon Lee
Abstract
A minimal surface is a surface that locally minimizes its area. We derive the first variation formula and give some important consequences such as the monotonicity formula and the mean value inequality. Then we derive the second variation formula and the stability inequality, and define the Morse index of a minimal surface.
Read the first variation formula(page 7), the monotonicity formula(proposition 1.12), the mean value inequality(proposition 1.15), and the second variation formula(page 40).
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