### 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).

**1. Wenesday, 19th, June**

*: Tran Thanh Hưng (Texas Tech.)*

**Speaker***: An Introduction to Singular Minimal Hypersurfaces*

**Title****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.

**2. Friday, 28th, June**

*Nguyen Thac Dung*

**Speaker:***First stability eigenvalues of singular minimal hypersurfaces in spheres*

**Title:****3. Wenesday, 3th, July**

*Nguyen Thac Dung*

**Speaker:***First stability eigenvalues of singular minimal hypersurfaces in spheres, II*

**Title:****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

**Wenesday, 10th, July**

*Tran Thanh Hung*

**Speaker:***A maximum principle for singular minimal hypersurfaces*

**Title:****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.

**Wenesday, 17th, July**

*Tran Thanh Hung*

**Speaker:***A maximum principle for singular minimal hypersurfaces II*

**Title:****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.

**Wenesday, 24th, July**

*Nguyễn Minh Hoàng*

**Speaker:***Stable minimal hypercones in Rn with n ≤ 7.*

**Title:****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:

**Wenesday, 7th, August**

*Nguyễn Minh Hoàng*

**Speaker:***Stable minimal hypercones in Rn with n ≤ 7.*

**Title:****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:

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**8. Wenesday, 14 ^{th}, 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 R

^{n+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, 21 ^{th}, 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 R

^{n+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, 20 ^{th}, 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.