Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.
These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions.
278-279, 289–290, 293, 365,423 Darboux, G. 52, 133, 135, 149, 194, 277 Davids, N. 365 De Giorgi, E. 86, 284 Dierkes, U. 292; 87,278, 424–426 Do Carmo, M. 48, 88 Dombrowski, P. 52, 80 Douglas, J. 9,340; 221, 277 Dubrovin, B. A. 48 Dziuk, ...
Existence and Regularity of Minimal Surfaces on Riemannian Manifolds
Regularity of Minimal Surfaces In this chapter we can finally prove that partial regularity of minimal surfaces; namely we show that the reduced boundary 6*E is analytic and the only possible singularities must occur in 6E – 6*E. Our ...
1985, D. Hoffman and W. Meeks, [HoMe1], proved that Costa's surface was embedded; this surface is now known as the Costa-Hoffman-Meeks surface. Moreover, Hoffman and Meeks showed that Costa's surface was just the first in a family of ...
The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations.
There is no comparably comprehensive treatment of the problem of boundary regularity of minimal surfaces available in book form. This long-awaited book is a timely and welcome addition to the mathematical literature.
Minimal hypercones and C*-minimizers for a singular variational problem. Indiana Univ. Math.J. 37,841-863 (1988) 6. ... Minimal immersions of spheres into spheres. Ann. Math.93, 43–62 (1971) Dombrowski, P. 1.
Federer and Fleming defined a k-dimensional surface in IR" as a k-current, i. e. a continuous linear functional on k-forms. Their method is treated in full detail in the splendid book of Federer [FH 1].
Minimal Surfaces: Boundary regularity