READING SEMINAR ON "MINIMAL HYPERSURFACES WITH A SINGULAR SET"
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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, 09:00 đến 10:30 ngày 24/07/2019, 09:00 đến 10:30 ngày 07/08/2019, 09:00 đến 10:30 ngày 20/08/2019, 14:00 đến 16:50 ngày 21/08/2019,

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
Abstract: The 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. 
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6. Wenesday, 24th, July
Speaker: Nguyễn Minh Hoàng
Title: Stable minimal hypercones in Rn with n ≤ 7.
Abstract: In this talk, we discuss minimal cones in Euclidean space. The study of these cones has been important both in the generalizations of the theorem of Bernstein and on issues of local regularity. The combined efforts of F.J.Almgren and J.Simons final gave the following theorem:
"The hyperplanes are the only stable minimal hypercones in Rn for n≤ 7".
The proof that we discuss here is due to Leon Simon.
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7. Wenesday, 7th, August
Speaker: Nguyễn Minh Hoàng
Title: Stable minimal hypercones in Rn with n ≤ 7.
Abstract: In this talk, we discuss minimal cones in Euclidean space. The study of these cones has been important both in the generalizations of the theorem of Bernstein and on issues of local regularity. The combined efforts of F.J.Almgren and J.Simons final gave the following theorem:
"The hyperplanes are the only stable minimal hypercones in Rn for n≤ 7".
The proof that we discuss here is due to Leon Simon.

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8. Wenesday, 14th, August

Speaker: Nguyen Thac Dung (VNU-HUS)

Title:  Entropy of closed hypersurfaces and singular self – shrinkers, part I

Abstract: In this talk, following the paper [1] by Zhu, I will prove a conjecture of Colding-Ilmanen-Minicozzi-White which is stated that any closed hypersurface in Rn+1 has entropy at least that of the round sphere, hold in any dimension n. The main ingredient of the proof is an extension of Colding-Minicozzi’s classification of entropy-stable self-shrinkers to the singular setting. This talk is based on the paper [1]

References

[1] J. J. Zhu, On the entropy of closed hypersurfaces and singular self-shrinkers, to appear in JDG.

[2] T. H. Colding and W. P. Minicozzi, II Generic mean curvature flow I: generic sungularies, Ann. of Math.  175 (2) 755 – 833.

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9. Wednesday, 21th, August.

Speaker: Nguyen Thac Dung (VNU-HUS)

Title:  Entropy of closed hypersurfaces and singular self – shrinkers, part II

Abstract: This is a continuation of the talk given last Wednesday. In this talk, following the paper [1] by Zhu, I will prove a conjecture of Colding-Ilmanen-Minicozzi-White which is stated that any closed hypersurface in Rn+1 has entropy at least that of the round sphere, hold in any dimension n. The main ingredient of the proof is an extension of Colding-Minicozzi’s classification of entropy-stable self-shrinkers to the singular setting. This talk is based on the paper [1].

References

[1] J. J. Zhu, On the entropy of closed hypersurfaces and singular self-shrinkers, to appear in JDG.

[2] T. H. Colding and W. P. Minicozzi, II Generic mean curvature flow I: generic sungularies, Ann. of Math.  175 (2) 755 – 833.

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10. Tuesday, 20th, 2019

Mini-workshop: SOME TOPICS ON GEOMETRIC ANALYSIS

Organizers: Le Minh Ha (VIASM), Ninh Van Thu (HUS), Nguyen Minh Hoang (HUS), Tran Thanh Hung (Texas Tech.)

Speakers: Juncheol Pyo (Pusan National University), Nguyen Thac Dung (Hanoi University Science), Nguyen Dang Tuyen (Hanoi University of Civil Engineering)

Program:

14h00-14h05: Welcome speech.

14h05-14h55: Juncheol Pyo

Title: Solitons for the mean curvature flow and inverse mean curvature flow

14h55-15h00: Tea break

15h00-15h30: Nguyen Dang Tuyen

Title: A Liouville theorem for a Lichnerowicz type equation on Riemannian manifolds. 

16h00-16h50: Nguyen Thac Dung

Title: Gradient estimates for a general heat equation under the Ricci flow

Abstracts:

Solitons for the mean curvature flow and inverse mean curvature flow - Juncheol Pyo (PNU)

Self-similar solutions and translating solitons are not only special solutions of mean curvature flow (MCF) but a key role in the study of singularities of MCF. They have received a lot of attention. We introduce some examples of self-similar solutions and translating solitons for the mean curvature flow (MCF) and give rigidity results of some of them. We also investigate self-similar solutions and translating solitons to the inverse mean curvature flow (IMCF) in Euclidean space. 

Gradient estimates for a general heat equation under the Ricci flow - Nguyen Thac Dung (HUS)

Given a complete, smooth metric measure space with the Bakry–Émery Ricci curvature bounded from below, various gradient estimates for solutions of general f-heat equations

are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases. This is a joint work with Khanh and Ngo.

A Liouville theorem for a Lichnerowicz type equation on weighted Riemannian manifolds. - Nguyen Dang Tuyen (NUCE)

In this talk, we consider a Lichnerowicz type equation on weighted Riemannian manifolds. Assume that a weighted Poincare inequality holds true, we prove a Liouville theorem for the equation. Our results are an improvement and a generalization of a recent work by Zhao Liang (appeared in JDE - 2019). This is a joint work in progress with Dung and Thoan.