| Assume (Q, xi) is a contact manifold with a contact one-form lambda. We assign an almost complex structure J compatible with (Q, lambda) and call (Q, lambda, J) a contact triad..;In this thesis, we define a contact instanton as a map from a Riemannan surface into a contact manifold, satisfying the following generalized Cauchy-Riemann typed equations 6&d1;p Jw=0,d&parl0;w*l&j0;j &parr0;=0. We derive a tensorial way for the analysis of contact instantons. The thesis mainly contains the following three contents.;First, we define the so called contact triad connection for each triad (Q, lambda, J) and prove the existence and uniqueness of such connection. This connection shows several advantages in the study the pseudo-holomorphic curves (contact instantons) in contact manifolds.;Second, we use the contact triad connection to study the analytic properties of contact instantons. In particular, we derive the energy density equality in a coordinate free form. Some new exploration of the tensorial calculations for the vector space valued forms are given for this context. Then we apply the Weitzenbock formula to the density equality, and derive coercive estimates for contact instantons from closed Riemann surfaces.;For a punctured Riemann surface, we study the asymptotical behavior of contact instantons defined on it. The subsequence convergence theorem is proven. We also apply the energy density equality under cylindrical coordinates and derive the exponential decay of the charge zero contact instantons to a limiting Reeb orbit under nondegenerate situation. An alternating boot-strapping method is presented in particular.;Third, we study the exponentially asymptotical behavior of contact instantons near a clean submanifold foliated by Reeb orbits which is of Morse-Bott setting (under some technical assumption for the clean submanifold). We give a normal form theorem to describe the tubular neighborhood of such clean submanifold. We give the splitting of contact instanton equations into vertical and horizontal parts with respect to the normal form. We prove the exponential decay of a contact instanton with vanishing charge to some Reeb orbit living in the clean submanifold. |
