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Properly Embedded Surfaces With Constant Mean Curvature Mojtaba Taherkhani m taherkhani@math iut ac ir Aug 2015 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Azam Etemad ae110mat@cc iut ac ir2000 MSC 53J60 53A10Keywords Maximum Principal H Surfaces Constant Mean Curvature Abstract In this thesis that besed on 21 we prove a maximum principle at infinity for properly embedded surfaces withconstant mean curvature H 0 in the 3 dimensional Euclidean space We show that no one of these surfaces can liein the mean convex side of another properly embedded H surface Maximum principles are used as basic analytictools for studying properties of functions defined on domains in Rn and satisfying certain equations e g elliptic Ingeneral these maximum principles play a fundamental role in analysis on complete Riemannian manifolds especiallyin the study of variational problems For example the well known maximum principle for harmonic functions has hadboth simplifying and unifying effect on the fields of harmonic and complex analysis The maximum principle is one ofthe most useful and best known tools employed in the study of partial differential equations The maximum principleenables us to obtain information about the uniqueness approximation boundedness and symmetry of the solutions Due to the local nature of maximum principle this principle can be used in Riemannian geometry for research ingeometry equations such as equations related to mean curvature As local nature of maximum principle it is notsenstive to the geometric properties of certain manifold M More than forty years ago in 1967 Omori duringthe study of immersions of Riemannian submanifolds in cones introduce the general state of maximum principlethat its origin is in the following simple observation If function u R R is bounded from above and we set 1 u supu then there exist a sequence xn R such that u xk u u xk 0 and u xk kfor every k N Omori introduce a form of maximum principle for Riemannian manifolds with sectional curvaturebounded from below and perhaps more important of that he also provide examples of manifolds that his generalform of maximum principle was failed Following the terminology introduced by Pigola Rigoli and Setti the Omori Yau maximum principle is said to hold on an n dimensional Riemannian manifold M if for any smooth functionu C 2 M with u supu there exists a sequence of points pk in M with the properties that 1 1 1u pk u u pk and u pk for every k N In this sense the classical k k kresult given by Omori 1967 and Yau 1975 states that the Omori Yau maximum principle holds on every completeRiemannian manifold with Ricci curvature bounded from below One problem in the theory of constant mean curvaturesurfaces cmc surfaces is to know when two surfaces with the same constant mean curvature can coexist in the sameambient space M More precisely if M1 and M2 are two properly immersed constant mean curvature surfaces in aRiemannian 3 manifold M these surfaces are called H surfaces then is the intersection M1 M2 empty If weconsider non intersecting properly immersed minimal surfaces in R3 D Hoffman and W Meeks proved that theseminimal surfaces are parallel planes For example any minimal surface on one side of a plane is a plane This resultis called the halfspace theorem This result can also be stated in another way Let us consider a properly immersedminimal surface in R3 with compact boundary and let P is a plane We assume that lies on one side of P thenthe distance between and P satisfies d P d P i e the distance is achieved along the boundary Sucha result is called a maximum principle at infinity In this thesis we prove that under natural assumptions if the 1surface lies in the slab x3 and is symmetric with respect to the plane x3 0 then it intersects this plane 2Hin a countable union of strictly convex closed curves

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