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.

 

Min-max theory for constant mean curvature hypersurfaces

Speaker: 

Jonathan Zhu

Institution: 

Harvard University

Time: 

Tuesday, January 16, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

We describe the construction of closed constant mean curvature (CMC) hypersurfaces using min-max methods. In particular, our theory allows us to show the existence of closed CMC hypersurfaces of any prescribed mean curvature in any closed Riemannian manifold. This work is joint with Xin Zhou.

Joint UCI-UCR-UCSD Southern California Differential Geometry Seminar

Institution: 

SCDGS

Time: 

Tuesday, February 20, 2018 - 3:00pm to 5:00pm

Location: 

AP&M 6402 UCSD

Speaker #1: Renato Bettiol (University of Pennsylvania) 

Title:  A Weitzenbock viewpoint on sectional curvature and application

Abstract:  In this talk, I will describe a new algebraic characterization of sectional curvature bounds that only involves curvature terms in the Weitzenboeck formulae for symmetric tensors. This characterization is further clarified by means of a symmetric analogue of the Kulkarni-Nomizu product, which renders it computationally amenable. Furthermore, a related application of the Bochner technique to closed 4-manifolds with indefinite intersection form and positive or nonnegative sectional curvature will be discussed, yielding some new nsight about the Hopf Conjecture. This is based on joint work with R. Mendes (Univ. Koln, Germany).

 

Speaker #2: Or Hershkovitzs (Stanford University)

Title: The topology of self-shrinkers and sharp entropy bounds

Abstract: The Gaussian entropy, introduced by Colding and Minicozzi, is a rigid motion and scaling invariant functional which measures the complexity of hypersurfaces of the Euclidean space. It is defined to be the supremal Gaussian area of all dilations and translations of the hypeprsurface, and as such, is well adapted to be studied by mean curvature flow. In the case of the n-th sphere in Rn+1, the entropy can be computed explicitly, and is decreasing as a function of the dimension n. A few years ago, Colding Ilmanen Minicozzi and White proved that all closed, smooth self-shrinking solutions of the MCF have larger entropy than the entropy of the n-th sphere. In this talk, I will describe a generalization of this result, which derives better (sharp) entropy bounds under topological constraints. More precisely, we show that if M is any closed self-shrinker in Rn+1 with a non-vanishing k-th homotopy group (with k less than or equal to n), then its entropy is higher than the entropy of the k-th sphere in Rk+1. This is a joint work with Brian White.

 

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