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The theory of minimal submanifolds is a fascinating field in differential geometry. The simplest, one-dimensional minimal submanifold, the geodesic, has been studied quite exhaustively, yet there are still a lot of interesting open problems.

Minimal surfaces. --- Minimal surfaces

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Differential geometry. Global analysis --- 514.1 --- Minimal surfaces --- Riemannian manifolds --- #WWIS:MEET --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Surfaces, Minimal --- Maxima and minima --- General geometry --- Minimal surfaces. --- Riemannian manifolds. --- 514.1 General geometry --- Geometrie differentielle globale --- Geometrie differentielle classique --- Surfaces minima

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"We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C Rn, without assuming any further regularity on the boundary of C. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of C, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension"--

Convex bodies. --- Boundary value problems. --- Isoperimetric inequalities. --- Calculus of variations and optimal control; optimization -- Manifolds -- Optimization of shapes other than minimal surfaces. --- Convex and discrete geometry -- General convexity -- Inequalities and extremum problems.

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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 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.

Boundary value problems. --- Minimal surfaces -- Data processing. --- Minimal surfaces. --- Minimal surfaces --- Boundary value problems --- Mathematics --- Civil & Environmental Engineering --- Geometry --- Calculus --- Operations Research --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Boundary conditions (Differential equations) --- Surfaces, Minimal --- Mathematics. --- Functions of complex variables. --- Partial differential equations. --- Differential geometry. --- Calculus of variations. --- Physics. --- Calculus of Variations and Optimal Control; Optimization. --- Differential Geometry. --- Partial Differential Equations. --- Functions of a Complex Variable. --- Theoretical, Mathematical and Computational Physics. --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Maxima and minima --- Mathematical optimization. --- Global differential geometry. --- Differential equations, partial. --- Complex variables --- Elliptic functions --- Functions of real variables --- Partial differential equations --- Geometry, Differential --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Mathematical physics. --- Physical mathematics --- Physics --- Differential geometry --- Isoperimetrical problems --- Variations, Calculus of

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One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementary proof of this very fundamental and beautiful mathematical result. The exposition follows the original line of attack initiated by Jesse Douglas in his Fields medal work in 1931, namely use Dirichlet's energy as opposed to area. Remarkably, the author shows how to calculate arbitrarily high orders of derivatives of Dirichlet's energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations, where normally only the second derivative or variation is calculated.  The monograph begins with easy examples leading to a proof in a large number of cases that can be presented in a graduate course in either manifolds or complex analysis. Thus this monograph requires only the most basic knowledge of analysis, complex analysis and topology and can therefore be read by almost anyone with a basic graduate education.

Global analysis. --- Mathematics. --- Minimal surfaces. --- Minimal surfaces --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Maxima and minima. --- Minima --- Surfaces, Minimal --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Sequences (Mathematics). --- Differential geometry. --- Functions of a Complex Variable. --- Sequences, Series, Summability. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Differential geometry --- Mathematical sequences --- Numerical sequences --- Algebra --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Maxima and minima --- Global differential geometry.