The Isoperimetric Inequality On Some Special Surfaces

The isoperimetric problem is one of the basic differential geometric problems.Its history is long.Many famous mathematicians once researched this problem.But most surfaces are not the plane in the reality.We can apply these method to more general problems by Researching the isoperimetric inequality on more general surfaces.This thesis discusses main works what I do during the period of graduate study.They include using the variational method to prove the circle among the curves of length L encloses the biggest area on the plane and the sphere and proving the isoperimetric inequality on compact convex surface in R~3.There are three chapters in this thesis.The first section of every chapter introduces some basic concepts.Chapter 1,this chapter discusses the isoperimetric problem on the plane.The first section introduces some basic concepts[9].The second section introduces Schmidt's method of proving isoperimetric inequality [2].The third section introduces Hurwitz's method [9].The fourth section takes from my tutor's lecture notes.The method is due to REILLY [10].Finally,I make use of variational method to prove the circle among the curves of length L encloses the biggest area on the plane.Chapter 2,this chapter introduces the isoperimetric problem on some special surfaces in R~3.Firstly,I introduce some basic concepts [3]and[9].Secondly,I use variational method to prove the circle among the curves of length L encloses the biggest area on the sphere.Finally,I prove the isoperimetric inequality on compact convex surface in R~3.Chapter 3,minimal surface is one of important surfaces in the differential geometry. In this chapter, I firstly deduce some conclusions on minimal surfaces. Then I introduce the isoperimetric inequality on minimal surfaces with connect boundary.