Abstract:
We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

Abstract:
We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of $CR$ structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''spherical space forms'' $\{\Gamma\backslash S^{3}\}$ as contact manifolds. We also propose the study of a certain kind of monopole equation associated with a contact structure. In view of the recently developed theory of contact homology algebras, we will discuss its overall impact on our study.

Abstract:
We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. This ``contact torsion'' is expected to be able to distinguish among ``contact lens'' spaces. We also propose the study of a certain kind of monopole equation associated with a contact structure.

Abstract:
We study the fillability (or embeddability) of $CR$ structures under the gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then it keeps having the same property after a short time. In the Appendix, we show the uniqueness of the solution to the gauge-fixed Cartan flow.

Abstract:
We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial $CR$ structure is fillable, then it keeps having the same property as long as the flow has a solution. We discuss the situation for the torsion flow and the Cartan flow. In the second part, we show that the above mentioned second order operator is used to express a tangency condition for the space of all fillable or embeddable $CR$ structures at one embedded in $\mathbb{C}^{2}.$

Abstract:
We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

Abstract:
We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We give an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

Abstract:
We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular domain$.$ By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are $C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.

Abstract:
We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ (on the nonsingular set) where $N_{\vec{F}}(w)$ $:=$ $\frac{\nabla w+\vec{F}}{|\nabla w+\vec{F}|}.$ To conclude $u$ $=$ $v$ (or $\nabla u$ $=$ $\nabla v)$, it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, for a generalized area functional including $p$-area, we prove that $N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ implies $\nabla u$ $=$ $\nabla v$ in dimension $\geq $ 3 under some rank condition on derivatives of $\vec{F}$ or the nonintegrability condition of contact form associated to $u$ or $v$. Note that in dimension 2 ($n=1),$ the above statement is no longer true. Inspired by an equation for the horizontal normal $N_{\vec{F}}(u),$ we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.

Abstract:
Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $% \Phi $ denote a $CR$ developing map {equation*} \Phi :\widetilde{M}\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$% . Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\Phi $ is injective for $n\geq 3$. In the case $n=2$, we also show that $\Phi $ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable.