*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Tuesday, June 18, 2019

2:10 p.m.

BA6183

PhD Candidate: Zhifei Zhu

Supervisor: Regina Rotman

Thesis title: Geometric inequalities on Riemannian manifolds

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In this thesis, we will show three results which partially answer several questions in the field of quantitative geometry. We first show that there exists Riemannian metric on a 3-disk so that the diameter, volume and surface area of the boundary is bounded, but during any contraction of the boundary of the disk, there exists a surface with arbitrarily large surface area. This result answers a question of P. Papasoglu.

The second result we will prove is that on any closed 4-dimensional simply-connected Riemannian manifolds with diameter <=D, volume > v > 0 and Ricci curvature |Ric| < 3, the length of a shortest closed geodesic can be bounded by some function f(v,D) which only depends on the volume and diameter of the manifold. This result partially answers a question of M. Gromov.

As an extension of our second result, we show that on 4-dimensional Riemannian manifolds satisfying the above conditions, the first homological filling function HF_1(l) <=f_1 (v,D)l +f_2 (v,D), for some functions f_1 and f_2 which only depends on v and D. And in particular, the area of a smallest minimal surface on the manifold can be bounded by some function which only depends on v and D.

A copy of the thesis can be found here: ut-thesis