# Research

## Differential Geometry

### Wednesday, November 28, 2018

Title: Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms
Speaker: Tian Chong
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

In this talk, we introduce some elliptic differential inequalities from the Weitzenbock formulae for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $$Lp$$ and $$L\infty$$ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among Kähler manifolds with constant scalar curvature.

### Wednesday, November 14, 2018

Title: Polarizable Carnot groups and Polar Coordinates
Speaker: Thomas Bieske
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

We will explore the geometric issues in establishing polar coordinates in Carnot groups. We will then discuss which groups have polar coordinates and which do not.

### Wednesday, November 7, 2018

Title: Solutions to $$p$$-Laplacian Equations with Drift Terms in Grushin-type Spaces
Speaker: Keller Blackwell
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

In this talk we introduce sub-Riemannian spaces called Grushin-type spaces. We will then explore past, present, and future work done in solving the $$p$$-Laplacian equation with drift terms in such spaces.

### Wednesday, October 31, 2018

Title: Equivalence of Weak and Viscosity solutions to the $$p(x)$$-Laplace Equation
Speaker: Robert Freeman
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

We discuss the differences between the $$p$$-Laplace equation and the $$p(x)$$-Laplace equation in Euclidean space and Carnot groups. We mention the different notions of solution to the $$p(x)$$-Laplace equation and outline the proof of the equivalence of the solutions in Carnot groups.

### Wednesday, October 24, 2018

Title: A $$\mathrm{p}(\cdot)$$-Poincaré Type Inequality and $$\mathrm{p}(\cdot)$$-Capacity in Carnot Groups, Part II
Speaker: Robert Freeman
Time: 2:00pm‐3:15pm
Place: CMC 108

### Wednesday, October 17, 2018

Title: A $$\mathrm{p}(\cdot)$$-Poincaré Type Inequality and $$\mathrm{p}(\cdot)$$-Capacity in Carnot Groups
Speaker: Robert Freeman
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

We discuss the $$\mathrm{p}(\cdot)$$-Capacity and quasi-continuity in the sub-Riemannian setting of Carnot groups. We will use the $$\mathrm{p}(\cdot)$$-capacity and quasi-continuity to show a $$\mathrm{p}(\cdot)$$-Poincaré-type inequality. We will then use this inequality to establish the existence of a minimizer to the $$\mathrm{p}(\cdot)$$-Dirichlet energy integral. Moreover, we use the $$\mathrm{p}(\cdot)$$-Poincaré-type inequality to show the uniqueness of the minimizer, up to a set of zero $$\mathrm{p}(\cdot)$$-capacity, in the Sobolev sense.

### Wednesday, October 10, 2018

Title: A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups, Part III
Speaker: Zachary Forrest
Time: 2:00pm‐3:15pm
Place: CMC 108

### Wednesday, October 2, 2018

Title: A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups, Part II
Speaker: Zachary Forrest
Time: 2:00pm‐3:15pm
Place: CMC 108

### Wednesday, September 26, 2018

Title: A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups
Speaker: Zachary Forrest
Time: 2:00pm‐3:15pm
Place: CMC 108

#### Abstract

We explore a Theorem for Euclidean spaces in the style of Radó for $$p$$-Harmonic functions as presented in a paper of Juutinen & Lindqvist, utilizing techniques of Viscosity Solutions. We consider the connection between the underlying Geometry and the proof of an essential Lemma in the paper, and speculate regarding the adaptability of this proof to the Heisenberg and (more generally) Carnot spaces.

### Wednesday, September 19, 2018

Title: Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part IV
Speaker: Diego Ricciotti
Time: 2:00pm‐2:50pm
Place: CMC 108

#### Abstract

We discuss De Giorgi classes and Holder continuity of the horizontal derivatives of p-harmonic functions in the Heisenberg group.

### Wednesday, September 12, 2018

Title: Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part III
Speaker: Diego Ricciotti
Time: 2:00pm‐2:50pm
Place: CMC 108

We present a proof of Lipschitz regularity for $$p$$-harmonic functions in the Heisenberg group for $$1 ### Wednesday, September 5, 2018 Title: Regularity of weak solutions of the \(p$$-Laplace equation in the Heisenberg group — Part II
Speaker: Diego Ricciotti
Time: 2:00pm‐2:50pm
Place: CMC 108

#### Abstract

Continuing from the regularity results presented last time, we will start discussing the nonlinear case ($$p$$ different than $$2$$).

### Wednesday, August 29, 2018

Title: Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part I
Speaker: Diego Ricciotti
Time: 2:00pm‐2:50pm
Place: CMC 108

#### Abstract

I will present some regularity results for weak solutions of the $$p$$-Laplace equation in the Heisenberg group based on difference quotients techniques, starting with the case $$p=2$$.