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.

 

Obstructions to the existence of conformally compact Einstein manifolds

Speaker: 

Matthew Gursky

Institution: 

University of Notre Dame

Time: 

Tuesday, March 13, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

In this talk I will describe a singular boundary value problem for Einstein metrics.  This problem arises in the Fefferman-Graham theory of conformal invariants, and in the AdS/CFT correspondence.   After giving a brief overview of some important results and examples, I will present a recent construction of boundary data which cannot admit a solution.  Finally, I will introduce a more general index-theoretic invariant which gives an obstruction to existence in the case of spin manifolds.  This is joint work with Q. Han and S. Stolz.

Convergence of Riemannian manifolds with scale invariant curvature bounds

Speaker: 

Norman Zergaenge

Institution: 

University of Warwick

Time: 

Tuesday, January 30, 2018 - 4:00pm

Host: 

Location: 

RH 306

A key challenge in Riemannian geometry is to find ``best" metrics on compact manifolds. To construct such metrics explicitly one is interested to know if approximation sequences contain subsequences that converge in some sense to a limit manifold.

In this talk we will present convergence results of sequences of closed Riemannian
4-manifolds with almost vanishing L2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls.  For instance we consider a sequence of closed Riemannian 4-manifolds,
whose L2-norm of the Riemannian curvature tensor is uniformly bounded from
above, and whose L2-norm of the traceless Ricci-tensor tends to zero.  Here,
under the assumption of a uniform non-collapsing bound, which is very close
to the euclidean situation, and a uniform diameter bound, we show that there
exists a subsequence which converges in the Gromov-Hausdor sense to an
Einstein manifold.

To prove these results, we use Jeffrey Streets' L2-curvature 
ow. In particular, we use his ``tubular averaging technique" in order to prove fine distance
estimates of this flow which only depend on significant geometric bounds.

Stable Horizons and the Penrose Conjecture

Speaker: 

Henri Roesch

Institution: 

UC Irvine

Time: 

Tuesday, March 6, 2018 - 4:00pm to 5:00pm

Location: 

RH 306

In the first half of the talk, we introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in spacetime or, equivalently, foliations of a null cone. We also show how certain, fairly generic, convexity assumptions on the null cone allows for a proof of the Penrose Conjecture. On the Black Hole Horizon, we find that the convexity assumptions become sharp; therefore, the second half of the talk will explore the existence of a class of Black Hole Horizons admitting such convexity. From this, building upon the work of S. Alexakis, we will show that the Schwarzschild Null Cone--the case of equality for the Penrose Conjecture--is also critical in light of recent work on the perturbation of stable, weakly isolated Horizons.

 

 

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