Global Gevrey Hypoellipticity For Twisted Laplacians

This thesis is a study on the global hypoellipticity in the Gevrey class for the systems of differential operator when the(anisotropic)twisted Laplacian acting on forms rather than the functions.First,using the rules of wedge and pull back com-putation in differential geometry,we get the representation of the differential systems of the twisted Laplacian acting on forms.Next,according to the definition of the Gevrey-hypoellipticity for operators acting on differential forms,in order to the Gevrey-hypoellipticity of twisted Laplacian acting on forms,we mainly have to prove that each coefficient of the differential forms is Gevrey-hypoelliptic.That is,each coefficient of the differential forms has to satisfy the similar a priori estimate shown in Proposition 3.1 in[41].So we introduce the parameter into the differential systems,and use the energy method and interconnection of the system to get the key a priori estimate for Gevrey-hypoellipticity by taking the parameter large enough.Finally,we verify that the twisted Laplacian acting on forms has the same Gevrey-hypoellipticity wether it has parameter or not.Thus we can extend the global Gevrey hypoellipticity of the twisted Laplacian acting on functions to the case that acting on forms.