Dimension Of Moduli Space Of Real Pseudoholomorphic Curves And Its Applications

In this thesis,we introduce the moduli space of real pseudoholomorphic curves and study its properties.We calculate virtual dimension of moduli space of real pseu-doholomorphic curves and establish some important inequalities in dimension 3.In the end we give possible applications of our results in the future.In chapter one and two,we first give an introduction to our main results and set up our conventions and notations.In chapter three,we give the complete definition of moduli space of real pseudo-holomorphic curves.And we calculate the virtual dimension of moduli space of real pseudoholomorphic curves with given boundary condition.The proof of the main re-sult consist of a series of lemmas,by using the pair of pants induction see[29].We calculate the index of Cauchy-Riemann operator on a disc first.And we prove gluing formula and Riemann-Roch formula with real symmetry on a close surface.Combin-ing results above,we can get the index of Cauchy-Riemann operator on an arbitrary Riemann surface with boundary and involution.Then we calculate the dimension of Teichm¨uller space and automorphism space of Riemann surface with involution.At last we can calculate the virtual dimension by definition.Especially when there is just one positive symmetric puncture,we have that the virtual dimension of moduli s-pace （?）.In chapter 4,we establish some inequalities of dimension of moduli space of real pseudoholomorphic curves in dimension 3.In first step,we give precise index iter-ation formulae for nondegenerate brake orbit in dimension 3,which consists of one elliptic type and four hyperbolic types.Based on index iteration formulae,we can prove the following three inequalities of dimension of moduli space of real pseudo-holomorphic curves in dimension 3.The first inequality is ind_R（u）≥0,where u is a real branched cover of a real trivial cylinder.Then we introduce real ECH index I_RECH and prove real ECH index is the upper bound of virtual dimension of moduli space of real pseudoholomorphic curves,i.e.ind_Ru≤I_RECH（β₁,β₂;Z）.The last inequality is ind_Ru≥Dind_R（（?））+（B+1+D）-#₁-#₂,where u is a real pseudoholomorphic curve,which is a real multiple cover of a somewhere injective real pseudoholomorphic curve（?）.In chapter five,we give possible applications of our above results.In the construc-tion of cylindrical contact homology for contact form of dynamical convex in dimen-sion 3,M.Hutchings and J.Nelson used index inequalities to ruled out all bad cases in boundary of moduli space.Therefore they can well define cylindrical contact homology contact form of dynamical convex in dimension 3.See[16].We can rule out many bad cases in boundary of moduli space of real pseudoholomorphic curves,using method similar to theirs.However there is still one bad cases left,which can not be ruled out by our index inequalities.In the future we can construct real contact homology in all odd dimensions and real embedded contact homology in dimension 3.These theories will have many appli-cations in Hamiltionian systems and contact geometry.