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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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Regularity of Minimal Surfaces (Hardcover, 2nd, rev. and enlarged ed. 2010)
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Regularity of Minimal Surfaces (Hardcover, 2nd, rev. and enlarged ed. 2010)
Series: Grundlehren der mathematischen Wissenschaften, 340
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Total price: R4,106
Discovery Miles: 41 060
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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 Plateaus 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 Plateaus problem have no
interior branch points.
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