We survey standard topics in elementary differential geometry and complex analysis to build up the necessary theory for studying applications of univalent harmonic function theory to minimal surfaces. We then proceed to consider convex combination harmonic mappings of the form f=sf_1+(1-s) f_2 and give conditions on when f lifts to a one-parameter family of minimal surfaces via the Weierstrauss-Enneper representation formula. Finally, we demand two minimal surfaces M and M' be locally isometric, formulate a system of partial differential equations modeling this constraint, and calculate their symmetry group. The group elements generate transformations that when applied to a prescribed harmonic mapping, lift to locally isometric minimal surfaces with varying graphs embedded in mathbb{R}^3.

The focus of this project consists of investigating when a ruled surface is a minimal surface. A minimal surface is a surface with zero mean curvature. In this project the basic terminology of differential geometry will be discussed including examples where the terminology will be applied to the different subjects of differential geometry. In addition the focus will be on a classical theorem of minimal surfaces referred to as the Plateau's Problem.

[pt] Nesse trabalho descrevemos superfícies mínimas mergulhadas em espaços produtos M x R, onde M = R2, S2 e H2 que são folheadas por geodésicas (superfícies regradas ) e curvas de curvatura constante de M (supefícies cíclicas ). Em R2xR, ou seja, em R3 vamos demonstrar que só existem duas superfícies mínimas cíclicas, que são o catenóide e o exemplo de Riemann. Em seguida caracterizamos as superfícies mínimas cíclicas em S2 x R que formam uma família a dois parâmetros e por fim exibimos três famílias de dois parâmetros de superfícies mínimas cíclicas em H2 x R. / [en] In this work we describe minimal surfaces embedded in product spaces M x R, where M = R2, S2 and H2 which are foliated by geodesics (ruled surfaces) and curves of M with constant curvature (cyclic surfaces). In R2 x R, i.e. R3, we shall prove that there exist only two minimal cyclic surfaces which are the catenoid and the Riemann example. Then we characterize minimal cyclic surfaces in S2 x R; they form a two-parameter family. Finally we exhibit three two-parameter families of minimal cyclic surfaces in H2 x R.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1910. / Published also as: Journal für die reine und angewandte Mathematik, Bd. 139, Heft 1. Vita.

In this thesis, we study a Bäcklund transformation for minimal surfaces - surfaces with vanishing mean curvature - transforming a given minimal surface into a possible infinity of new ones. The transformation, also carrying with it mappings between solutions to the elliptic Liouville equation, is first derived by using geometrical concepts, and then by using algebraic methods alone - the latter we have not been able to find elsewhere. We end by exploiting the transformation in an example, transforming the catenoid into a family of new minimal surfaces.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1997. / Includes bibliographical references (p. 133-134).

Orientador: Francesco Mercuri / Dissertação (mestrado) - Universidade Estadual de Campinas, Intituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T13:33:06Z (GMT). No. of bitstreams: 1 Min_LienKuan_M.pdf: 1157875 bytes, checksum: 65f63453a02a7c1365c0a9b3524a1602 (MD5) Previous issue date: 2006 / Resumo: O teorema de Bernstein é um marco importante na teoria das superfícies mínimas. Nesta dissertação apresentaremos três demonstrações deste teorema, cada uma levando a generalizações em diferentes direções / Abstract: The Bernstein's theorem is an important landmark in the theory of the minimal surfaces. In this dissertation we will present three demonstrations of this theorem, each one leading to generalizations in different directions / Mestrado / Geometria Diferencial / Mestre em Matemática

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Obtemos limites inferiores para o tom fundamental de conjuntos abertos em subvariedades com curvatura mÃdia localmente limitada no espaÃo produto N x R, onde N à uma variedade Riemanniana completa n-dimensional com curvatura seccional K à menor ou igual que a curvatura do espaÃo forma. Quando a imersÃo à mÃnima nossas estimativas sÃo Ãtimas. / We give lower bounds for the fundamental of open sets in submanifolds with locally bounded mean curvature in N X R, where N is an n-dimensional complete Riemannian manifold with radial sectional curvature KN is less than or equal to the curvature of space form. When the immersion is minimal our estimates are sharp.

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / O presente trabalho de mestrado visa estudar alguns dos trabalhos do matemÃtico italiano Ennio De Giorgi os quais fazem referÃncia a existÃncia e regularidade de superfÃcies mÃnimas mas estas nÃo contextualizadas integralmente no Ãmbito da Geometria Diferencial mas sim voltadas a um campo da matemÃtica a algumas dÃcadas implementada que a Teoria GeomÃtrica da Medida. Segundo as definiÃÃes de Ennio De Giorgi iremos estudar superfÃcies, que para o mesmo se davam como bordos de certos conjuntos, os quais sÃo denotados de conjuntos de Caccioppoli, homenagem esta dada por Di Giorgi a o matemÃtico italiano Renato Caccioppoli. Tais conjuntos tem muitas propriedades geomÃtricas interessantes, como por exemplo adimetem plano tangente canÃnico em quase todo ponto, e, possuem âperÃmetroâ finito. Os resultados expostos constatarÃo que atà a dimensÃo 7 todas as soluÃÃes do problema de Plateau sÃo regulares e em geral sua classe de regularidade à C1,&#945;. Enfim, os resultados deste trabalho em sua maioria serÃo baseados na obra: Minimal Surface and Function of Bounded Variation do autor Enrico Giusti, o qual resume bem as tÃcnicas de Teoria GeomÃtrica da Medida referentes aos trabalhos de Ennio De Girogi sobre teoria de regularidade de superfÃcies mÃnimas. Teorema (De Giorgi-Federer-Massari-Miranda). Sejam &#937; contido em R^n, n >1 um conjunto aberto e E contido em R^n um conjunto de Caccioppoli satisfazendo para &#945; em (0, 1) &#968;(E, B&#961;(x)) < c&#961;^(n-1+2&#945;) para todo x em &#937; e todo &#961; em (0, R), com c e R constantes positivas. EntÃo a fronteira reduzida à uma hipersuperfÃcie analÃtica C1,&#945; em &#937;, e H^s((&#8706;E - &#8706;E) &#937;) = 0 para todo s > n - 8. AlÃm disso, suponha que Ej à uma sequÃncia de conjuntos minimais em B1 convergindo localmente a um conjunto mÃnimal C. Sejam x em &#8706;C e xj em Ej , xj convergindo a x. EntÃo, se j à suficientemente grande, xj à um ponto regular para &#8706;Ej e &#957;Ej(xj) converge a &#957;(x), onde &#957;(x) à o vetor normal relativo a Ej, &#8706;E denota a fronteira reduzida de E e H^s denota a medida de Hausdoff. / This master thesis aims to study some of the work of the Italian mathematician Ennio De Giorgi which refer to the existence and regularity of minimal surfaces but they do not fully contextualized within the framework of differential geometry but rather focused on a field of mathematics that implemented a few decades Geometric Measure Theory. According to the definitions of Ennio De Giorgi will study surfaces, which gave to the same as certain boundarys of sets, which are denoted sets Caccioppoli, this honor given by De Giorgi the Italian mathematician Renato Caccioppoli. These sets have many interesting geometric properties, such as adimits canonical tangent plan almost everywhere, and have "perimeter" finite. The above results will see that even the size 7 all the solutions to the problem of Plateau are regular and in general their regular class is C1,&#945;. Finally, the results of this study are mostly based on the work: Minimal Surface and Function of Bounded Variation of the author Enrico Giusti, which summarizes the techniques of Geometric Measure Theory relating to the work of Ennio De Girogi on a regularity theory of minimal surfaces Theorem (De Giorgi- Federer- Massari - Miranda) Let &#937; contains in R^n, n>1 an open set and E contains in R^n a Caccioppoli set satisties to &#945; in (0, 1) &#968;(E, B&#961;(x)) < c&#961;^(n-1+2&#945;) for every x in &#937; and every &#961; in (0, R), with c and R positive constants. Then the reduced boundary is a hipersurface analitic C1,&#945; in &#937;, e H^s((&#8706;E - &#8706;E) &#937;) = 0 for every s > n - 8. Furthemore, suppose that Ej is a sequence of minimal sets in B1 locally converging at minimal set C. Let x in &#8706;C and xj in Ej , xj converging at x. Then, if j is enough large , xj is a regular point for &#8706;Ej and &#957;Ej(xj) converges to &#957;(x), where &#957;(x) is a normal vector relative at Ej, &#8706;E denots the boundary reduced of E and H^s denots the Hausdorff measure.

Let f_1, f_2 be univalent harmonic mappings of some planar domain D into the complex plane C. This thesis contains results concerning conditions under which the convolution f_1 ∗ f_2 or the convex combination tf_1 + (1 − t)f_2 is univalent. This is a long-standing problem, and I provide several partial solutions. I also include applications to minimal surfaces.