The Plateau’s Problem With Algebraic Boundary Conditions And Functionalized Ulam Floating Body Problem

The affine surface area is a basic affine invariant in affine differential geometry,and it is a basic tool for the study of the approximation theory of convex bodies and valuations theory.In the early 1990s,the notion of affine surface area was generalized to arbitrary convex surface,and these generalizations could be used to prove properties such as the upper semicontinuity conjecture and valuations of affine surface area.These properties play an important role in dealing with geometric problems,among which,the upper semicontinuity of affine surface area is proved to be the key to solving the affine Plateau’s problem.Now,scholars have found that the affine surface area is also closely related to the Ulam’s longstanding problem,where the Ulam floating body can be used to obtain the affine surface area.The dissertation is devoted to the study of the Plateau’s problem with algebraic boundary conditions and functionalized Ulam floating body problem,which are two types of hot issues in geometric field,including the problem of varifolds generated by rectifiable chains,the Plateau’s problem with algebraic boundary characterized by Cech homology groups,minimal currents problem,minimal flat chains problem,and the functionalized Ulam floating body problem.The main work of this paper is as follows:Chapter 3 studies the varifolds generated by rectifiable chains.Firstly,we consider that the coefficient group is a complete normed abelian group,give the definition of rectifiable varifolds generated by rectifiable chains.Nextly,it is proved that the convergence of polyhedral chains in flat norm can deduce the weak convergence of the generated rectifiable varifolds under proper conditions.Finally,the conclusion on polyhedral chains is generalized to rectifiable chains.Chapter 4 studies the Plateau’s problem with homological algebraic boundary.Firstly,we consider that the coefficient group is an abelian group,and prove that if the homologies H*satisfy the proper axioms conditions,the infimum of the Plateau’s problem with algebraic boundary characterized by Cech homology groups is equal to the infimum of the Plateau’s problem with algebraic boundary characterized by H*homology groups.Nextly,we consider that the coefficient group is a discrete normed abelian group,construct the isomorphism between the chain homology groups and the Cech homology groups,and prove that the infimum of the Plateau’s problem with algebraic boundary characterized by Cech homology groups is equal to the infimum of size minimizing flat chains problem.Finally,we consider that the coefficient group is an integer group,construct the isometric isomorphism between the chain groups and the corresponding current groups,and prove that the infimum of the Plateau’s problem with algebraic boundary characterized by Cech homology groups is equal to the infimum of size minimizing currents problem.Chapter 5 studies the functionalized Ulam floating body problem.Firstly,we consider the special closed convex set such as the epigraph of a convex function,and give the definitions of the Ulam floating functions for convex and log-concave functions.Nextly,it is proved that the affine-invariance of Ulam floating functions.Finally,a close connection between Ulam floating functions and affine surface area for log-concave functions is established.