The Morse index of Free Boundary Minimal Hypersurfaces

Speaker: 

Hung Tran

Institution: 

Texas Tech University

Time: 

Tuesday, May 22, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

A free boundary minimal hypersurface in the unit Euclidean ball is a critical point of the area functional among all hypersurfaces with boundaries in the unit sphere, the boundary of the ball. While regularity and existence aspects of this subjecct have been extensively investigated, little is known about uniqueness. That motivates the study of the Morse index, which quantitatively measures the number of deformations decreasing the area to second order. Henceforth, A. Fraser and R. Schoen proposed a fundamental conjecture concerning surfaces with low indices. In this talk, we discuss recent developments including a joint work with Ari Stern, Detang Zhou, and Graham Smith.

ALE Kahler Manifolds

Speaker: 

Rares Rasdeaconu

Institution: 

Vanderbilt

Time: 

Tuesday, June 5, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

In recent years the scalar flat asymptotically locally Euclidean (ALE) Kahler manifolds attracted a lot of attention, and many examples were constructed. However, their classification is not understood, except for the case of ALE Ricci flat Kahler surfaces. In this talk, I will present a first step in this direction: the underlying complex structure of ALE Kahler manifolds is exposed to be a resolution of a deformation of an isolated quotient singularity. The talk is based on a joint work with Hans-Joachim Hein and Ioana Suvaina.

Kahler-Ricci solitons on crepant resolutions of quotients of C^n

Speaker: 

Heather Macbeth

Institution: 

MIT

Time: 

Tuesday, April 3, 2018 - 4:00pm to 5:00pm

Location: 

RH 306

By a gluing construction, we produce steady Kahler-Ricci solitons on equivariant crepant resolutions of C^n/G, where G is a finite subgroup of SU(n), generalizing Cao's construction of such a soliton on a resolution of C^n/Z_n.  This is joint work with Olivier Biquard.

Compactness and generic finiteness for free boundary minimal hypersurfaces

Speaker: 

Qiang Guang

Institution: 

UC Santa Barbara

Time: 

Tuesday, May 1, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Free boundary minimal hypersurfaces are critical points of the area functional in compact manifolds with boundary. In general, a free boundary minimal hypersurface may be improper, i.e., the interior of the hypersurface may touch the boundary of the ambient manifold. In this talk, we will present recent work on compactness and generic finiteness results for improper free boundary minimal hypersurfaces. This is joint work with Xin Zhou. 

Joint UCI-UCR-UCSD Southern California Differential Geometry Seminar

Institution: 

SCDGS

Time: 

Tuesday, May 8, 2018 - 3:00pm to 5:00pm

Location: 

UC Riverside

Lecture 1

Speaker: Otis Chodosh

Time/place: Surge 284 3:40~4:30

Title:Properties of Allen--Cahn min-max constructions on 3-manifolds

Abstract:

I will describe recent joint work with C. Mantoulidis in which we study the properties of bounded Morse index solutions to the Allen--Cahn equation on 3-manifolds. One consequence of our work is that a generic Riemannian 3-manifold contains an embedded minimal surface with Morse index p, for each positive integer p.

 

Lecture 2

Speaker:  Ved Datar

Time/place: Surge 284 4:40~5:30

Title: Hermitian-Yang-Mills connections on collapsing K3 surfaces

Abstract:

Let $X$ be an elliptically fibered K3 surface with a fixed $SU(n)$ bundle $\mathcal{E}$. I will talk about degenerations of connections on $\mathcal{E}$ that are Hermitian-Yang-Mills with respect to a collapsing family of Ricci flat metrics. This can be thought of as a vector bundle analog of the degeneration of Ricci flat metrics studied by Gross-Wilson and Gross-Tosatti-Zhang. I will show that under some mild conditions on the bundle, the restriction of the connections to a generic elliptic fiber converges to a flat connection. I will also talk about some ongoing work on strengthening this result. This is based on joint work with Adam Jacob and Yuguang Zhang.

 

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