READING SEMINAR ON "MINIMAL HYPERSURFACES WITH A SINGULAR SET"

Time: 09:00 đến 10:30 ngày 19/06/2019, 09:00 đến 10:30 ngày 28/06/2019, 09:00 đến 10:30 ngày 03/07/2019, 09:00 đến 10:30 ngày 10/07/2019, 09:00 đến 10:30 ngày 17/07/2019,

Venue/Location: C2-714, VIASM

Abstract: The goal of this seminar is to understand how to extend theorems from the regular setting to the singular one. The context of most interests is about minimal surfaces possibly with some boundary conditions (preferably free).
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The program is as follows.
1. Wenesday, 19th, June
Speaker: Tran Thanh Hưng (Texas Tech.)
Title: An Introduction to Singular Minimal Hypersurfaces
AbstractThe goal of this reading seminar is to learn how to extend results from the regular setting to a singular one in the context of minimal hypersurfaces. In this introductory talk, we first briefly recall the theory of minimal surfaces in the three-dimensional Euclidean space. Here singularities might arise as branch points which can be understood using complex analysis. Then, utilizing the language of geometric measure theory, we'll describe singular minimal hypersurfaces in any dimension. Here, the emphasis will be on the intuition rather than technicalities. 
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2. Friday, 28th, June
Speaker: Nguyen Thac Dung
Title: First stability eigenvalues of singular minimal hypersurfaces in spheres
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3. Wenesday, 3th, July
Speaker: Nguyen Thac Dung
Title: First stability eigenvalues of singular minimal hypersurfaces in spheres, II
Abstract: In this talk, I report the paper by Zhu mentioned in the title. Following the paper, I give an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, I show that any singular minimal hypersurface in Sn+1, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability eigenvalue at most -2n, with equality if and only if it is a product of two round spheres.
Reference: J. J. Zhu, First stability eigenvalues of singular minimal hypersurfaces in spheres,  Cacl. Var. (2018) 57:130 
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4. Wenesday, 10th, July
Speaker: Tran Thanh Hung
Title: A maximum principle for singular minimal hypersurfaces
Abstract: Consider two minimal hypersurfaces with a common point and, locally around it, one hypersurface lies on one side of the other. If both surfaces are smooth,  a well-known consequence of the elliptic maximum principle implies that they must coincide. In this talk, following the work of L. Simon (87), T. Ilmanen (96), and N. Wickramasekera (14), we discuss how to extend the result to the singular setting, namely when the common point is singular. The approach is to understand the tangent cones at singular points. 
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5. Wenesday, 17th, July
Speaker: Tran Thanh Hung
Title: A maximum principle for singular minimal hypersurfaces II
Abstract: Consider two minimal hypersurfaces with a common point and, locally around it, one hypersurface lies on one side of the other. If both surfaces are smooth,  a well-known consequence of the elliptic maximum principle implies that they must coincide. In this talk, following the work of L. Simon (87), T. Ilmanen (96), and N. Wickramasekera (14), we discuss how to extend the result to the singular setting, namely when the common point is singular. The approach is to understand the tangent cones at singular points.