京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18444号 / 理博第4004号 / 新制||理||1577(附属図書館) / 31322 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 堤 誉志雄, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Die hier dargelegte Dissertation ist motiviert durch die Vorschläge von Joyce, Doan und Walpuski zur Definitionen enumerativer Invarianten für G2-Mannigfaltigkeit, durch das Zählen gewisser kalibrierter Untermannigfaltigkeiten, sogenannter assoziativen Untermannigfaltigkeiten. In Kapitel 1, werde ich Definitionen und grundlegende Fakten über G2-Mannigfaltigkeit und deren assoziative Untermannigfaltigkeit wiederholen. Darüber hinaus erläutere ich die Konstruktion von G2-Mannigfaltigkeit als verdrehte verbundener Summe. Kapitel 2 schafft die nötige Grundlage für das darauf folgende dritte Kapitel. Hier definiere ich den Modul-Raum der asymptotisch zylindrischen assoziativen Untermannigfaltigkeiten zusammen mit seiner natürlichen Topologie und zeige, dass der Modul-Raum lokal homeomorph zur Urbild-Menge der Null einer glatten Abbildung zwischen zwei endlich-dimensionalen Räu- men ist. In besonderen Fällen ist dieser Modul-Raum eine Lagrangesche Untermannigfaltigkeit des Modul Raums der holomorphen Kurven einer asymptotisch zylindrischen Calabi-Yau Man- nigfaltigkeit. In Kapitel 3 beweise ich ein Klebe-Theorem für ein Paar von asymptotisch zylindrischen as- soziativen Untermannigfaltigkeiten in einem zusammenpassenden Paar von asymptotisch zylin- drischen G2-Mannigfaltigkeiten. Hiermit konstruiere ich neue geschlossene und starre (rigid) assoziative Untermannigfaltigkeiten in verdrehten verbundenen Summe G2-Mannigfaltigkeiten. In Kapitel 4 untersuche ich den Modul-Raum der konisch singulären assoziativen Un- termannigfaltigkeiten in G2-Mannigfaltigkeiten. Durch das Umformulieren des Indexes des Operators, der die Deformationstheorie kontrolliert, in bestimmte Stabilität-Indizes des zu- grundeliegenden assoziativen Kegels begründe ich, dass in einem generischen Pfad in dem Raum der ko-geschlossenen G2-Strukturen keine asymptotisch konischen assoziative Unter- mannigfaltigkeiten existieren, die mindestens eine Singularität besitzen, die auf einem Kegel mit Stabiltätsindex größer als eins modeliert werden. Dieses Resultat lässt sich auf alle speziellen Lagrangesche-Kegel außer den Harvey-Lawson-T2-Kegel und die Vereinigung zweier speziellen Lagrangesche-Flächen anwenden. Zusätzlich lässt sich das Ergebnis auch auf alle konischen assoziativen Untermannigfaltigkeiten anwenden, deren zugrundeliegende Verschlingung (link) holomorphe Kurven mit Null-Torsion in S6 sind. Des Weiteren dienen Teile des vierten Kapitels als Grundlage für das darauf folgende Kapitel 5. Aufgrund einiger Übergangsphänomene entlang eines generischen Pfades von G2-Strukturen, führt das naive Zählen von assoziativen Untermannigfaltigkeiten zu keiner Invariante. Tat- sächlich wurde vermutet, dass a) eine assoziative Untermannigfaltigkeit aus einer assoziativen Untermannigfaltigkeit mit Selbstsschnitt (self-intersection) geboren werden kann, und, dass b) drei assoziative Untermannigfaltigkeiten aus einer konisch singulären assoziativen Un- termannigfaltigkeit, deren Singularität durch den Harvey-Lawson-T2-Kegel modelliert wird, entspringen. In Kapitel 5, beweise ich ein Desingularitätstheorem für konisch singulären assoziative Untermannigfaltigkeit entlang eines Pfades von ko-geschlossenen G2-Strukturen. Somit verifiziere ich Vermutung b) bewiesen und teilweise auch Vermutung a). / The dissertation presented here is motivated from the proposals made by Joyce, Doan and Walpuski to define enumerative invariants of G2-manifolds by counting certain calibrated submanifolds, called associative submanifolds. In Chapter 1, I review the definitions and basic facts of G2-manifolds and associative submanifolds. Moreover, I explain the construction of G2-manifolds as twisted connected sums. Chapter 2 serves as a necessary groundwork for Chapter 3. Here, I define the moduli space of asymptotically cylindrical associative submanifolds with its natural topology and prove that the moduli space is locally homeomorphic to the zero set of a smooth map between two finite-dimensional spaces. In the best scenario, this moduli space is a Lagrangian submanifold of the moduli space of holomorphic curves in the asymptotic Calabi-Yau 3-fold. In Chapter 3, I prove a gluing theorem for a pair of asymptotically cylindrical associative submanifolds in a matching pair of asymptotically cylindrical G2-manifolds. Using this I construct new closed and rigid associative submanifolds of twisted connected sum G2-manifolds. In Chapter 4, I study the moduli space of conically singular associative submanifolds in G2-manifolds. By reformulating the index of the operator that controls the deformation theory in terms of certain stability-index of the associative cones, I establish that in a generic path of co-closed G2-structures there are no conically singular associative submanifolds that have at least one singularity modeled on a cone of stability-index greater than one. This result applies to all special Lagrangian cones, except the Harvey-Lawson T2-cone and a union of two special Lagrangian planes. Additionally, it applies to all associative cones whose links are null-torsion holomorphic curves in S6. Furthermore, parts of Chapter 4 also serve as a necessary groundwork for Chapter 5. The naive counting of associative submanifolds does not lead to an invariant due to several transitions that may occur along a generic path of G2-structures. In fact it was conjectured that a) an associative submanifold born out of an associative submanifold with self intersection, and b) three associative submanifolds arise from a conically singular associative submanifold whose singularity is modeled on Harvey-Lawson T2-cone. In Chapter 5, I prove a desingularization theorem for conically singular associative submanifolds along a path of co-closed G2-structures. Consequently, I verify conjecture b) and partially confirm conjecture a).

Machine strength grading of structural timber is a sawmill process by which considerable value is added to sawn products. The principle of such grading is that the strength of a timber member is predicted on the basis of a so called indicating property (IP) which, in general, represents an averaged value of the modulus of elasticity (MOE) measured over a board length of about one meter or more. A limitation of today’s grading methods is that the accuracy of strength predictions is often rather poor, which results in a low degree of utilization as regards structural potential of sawn timber. However, it has for many years been well known to researchers that much better strength predictions can be made by using localized MOE values, determined over a very short length, as IP. Still, the determination of such values in a sawmill production environment has been technically very difficult to achieve. In the research presented in this thesis, dot laser scanning with high resolution was utilized for detection of local fibre orientation on the surfaces of timber members. Since wood is an orthotropic material with superior structural performance in the longitudinal fibre direction, information about fibre orientation was, in combination with beam theory and measured wood material properties, used to determine the bending MOE variation along boards. By application of an IP defined as the lowest MOE found along a board, more accurate strength predictions than what is obtained by common commercial grading techniques was attained. The thesis also involves flatwise wet gluing of Norway spruce side boards into laminated beams. As side boards, being cut from the outer parts of a log, have excellent structural properties it was not surprising to find that the beams had high strength and stiffness, even when laminations of sawfalling quality were used. The possibility of grading boards in a wet state by means of axial dynamic excitation was investigated with a positive result and application of simple grading rules resulted in considerable improvement of beam bending strength. Finally, bending MOE variation determined on the basis of laser scanned fibre directions was used for identification of weak sections in laminations. Elimination of such sections by means of finger jointing showed that average lamination strength of a board sample could be improved by more than 35 percent. / Hållfasthetssortering av konstruktionsvirke är en sågverksprocess som innebär att värdet av det sågade virket ökar väsentligt. Principen för denna typ av sortering är att styrkan hos ett virkesstycke predikteras med utgångspunkt från en så kallad indikerande egenskap (IP) som oftast representeras av medelvärdet av elasticitetsmodulen (E-modulen) mätt över en sträcka av minst en meter utmed virkesstyckets längd. De sorteringsmetoder som används idag ger ofta prediktioner med relativt låg noggrannhet, vilket innebär att endast en begränsad del av det sågade virkets konstruktiva potential kan utnyttjas. Det är dock väl känt att avsevärt bättre prediktioner kan erhållas genom att använda en lokal E-modul, uppmätt över en mycket kort sträcka, som IP. I dagsläget saknas dock teknik för att kunna bestämma ett sådant lokalt värde vid produktionshastighet i ett sågverk. I den forskning som presenteras i denna avhandling har punktlaserskanning med hög upplösning använts för att bestämma fiberriktningens variation på ytorna av virkesstycken. Eftersom trä är ett ortotropt material med högst styvhet och styrka i longitudinell fiberriktning ger skanningsresultaten värdefull information om hur dessa egenskaper varierar längs en planka. Genom att kombinera informationen om fiberriktning med uppmätta virkesegenskaper och klassisk balkteori, kan böjstyvhetens variation utmed en planka beräknas med hög upplösning och därefter omräknas till en E-modul i böjning. Med en IP definierad som det lägsta värdet på nämnda E-modul utmed en planka kan en högre noggrannhet i prediktionen av hållfasthet uppnås, jämfört med vad som kan erhållas med dagens sorteringsmetoder. Avhandlingen omfattar också limträbalkar tillverkade av sidobräder av gran limmade i rått tillstånd. Eftersom sidobräder sågas från de yttre delarna av en stock har de vanligtvis utmärkta konstruktiva egenskaper. Det var därför inte förvånande att balkarna uppvisade hög styrka och styvhet, även i de fall lamellerna var av sågfallande kvalitet. Möjligheten att med hjälp av axiell dynamisk excitering sortera sidobräder i rått tillstånd undersöktes med positivt resultat och genom att använda sorterade lameller kunde balkarnas styrka förbättras avsevärt. Den ovan beskrivna metoden att med utgångspunkt från bl.a. skannade fibervinklar bestämma styvhetens variation längs virkesstycken utnyttjades sedan för att identifiera svaga snitt i lameller av sidobräder. Genom att eliminera sådana snitt med hjälp av fingerskarvning kunde medelhållfastheten för ett stickprov sidobrädor höjas med mer än 35 %.

Este trabalho teve como finalidade o estudo da aplicação de um adesivo alternativo para a fabricação de madeira laminada colada (MLC). Este adesivo, à base de óleo de mamona, apresenta vantagens em relação ao tradicionalmente utilizado, considerando aspectos ecológicos e econômicos. Foram determinados parâmetros adequados para a colagem, e avaliada a compatibilidade entre adesivo e o tratamento com o preservativo hidrossolúvel de Arseniato de Cobre Cromatado (CCA - tipo C) e em madeiras sem o tratamento preservativo, em termos de resistência da linha da cola, por meio dos ensaios de cisalhamento, de tração normal e de tração paralela às fibras, usando a espécie Pinus caribea hondurensis. Com estes parâmetros, foram confeccionadas 12 vigas de MLC, utilizando-se as espécies de reflorestamento Pinus caribea hondurensis e Eucaliptus grandis. Foi avaliado o desempenho estrutural das vigas de MLC, por intermédio do ensaio mecânico de resistência à flexão, objetivando-se analisar a resistência da linha de cola. Os resultados obtidos permitem concluir o bom desempenho do adesivo poliuretano à base de mamona, para a aplicação em madeira não preservada e preservada com OCA tipo-C. A pressão de colagem indicada para laminação das vigas é 0,8% MPa para madeira não tratada, e 1,6 MPa para madeira tratada. Para a fabricação das emendas das lâminas recomenda-se a pressão de colagem igual a MPa, para madeira tratada ou não. / The aim of this work is the study of the use of alternative adhesive for glued laminated timber (GLULAM). This adhesive, based on castor oil, presents ecological and economical advantages in relation of traditionally used. The appropriate parameters for gluing were determined and the compatibility between this adhesive and treatment with the waterbome preservative of chromated copper arsenate (CCA-C) were evaluated through shear tests and tensions tests (perpendicular and parallel to the grain), using the species Pinus caribea hondurensis. With these parameters, were made 12 beams of GLULAM; using the species Pinus caribea hondurensis and Eucaliptus grandis. The structural performance of the beams of GLULAM was evaluated through static bending tests. The results obtained allow concluding the good performance of the adhesive polyurethane based on castor oil, for use in wood not preserved and preserved with CCA type-C. The gluing pressures indicated are 0,8 MPa for non-treated wood, and 1,6 MPa for treated wood. The gluing pressure recommended for finger-joints is 9 MPa, for treated wood or not.

Este trabalho teve como finalidade o estudo da aplicação de um adesivo alternativo para a fabricação de madeira laminada colada (MLC). Este adesivo, à base de óleo de mamona, apresenta vantagens em relação ao tradicionalmente utilizado, considerando aspectos ecológicos e econômicos. Foram determinados parâmetros adequados para a colagem, e avaliada a compatibilidade entre adesivo e o tratamento com o preservativo hidrossolúvel de Arseniato de Cobre Cromatado (CCA - tipo C) e em madeiras sem o tratamento preservativo, em termos de resistência da linha da cola, por meio dos ensaios de cisalhamento, de tração normal e de tração paralela às fibras, usando a espécie Pinus caribea hondurensis. Com estes parâmetros, foram confeccionadas 12 vigas de MLC, utilizando-se as espécies de reflorestamento Pinus caribea hondurensis e Eucaliptus grandis. Foi avaliado o desempenho estrutural das vigas de MLC, por intermédio do ensaio mecânico de resistência à flexão, objetivando-se analisar a resistência da linha de cola. Os resultados obtidos permitem concluir o bom desempenho do adesivo poliuretano à base de mamona, para a aplicação em madeira não preservada e preservada com OCA tipo-C. A pressão de colagem indicada para laminação das vigas é 0,8% MPa para madeira não tratada, e 1,6 MPa para madeira tratada. Para a fabricação das emendas das lâminas recomenda-se a pressão de colagem igual a MPa, para madeira tratada ou não. / The aim of this work is the study of the use of alternative adhesive for glued laminated timber (GLULAM). This adhesive, based on castor oil, presents ecological and economical advantages in relation of traditionally used. The appropriate parameters for gluing were determined and the compatibility between this adhesive and treatment with the waterbome preservative of chromated copper arsenate (CCA-C) were evaluated through shear tests and tensions tests (perpendicular and parallel to the grain), using the species Pinus caribea hondurensis. With these parameters, were made 12 beams of GLULAM; using the species Pinus caribea hondurensis and Eucaliptus grandis. The structural performance of the beams of GLULAM was evaluated through static bending tests. The results obtained allow concluding the good performance of the adhesive polyurethane based on castor oil, for use in wood not preserved and preserved with CCA type-C. The gluing pressures indicated are 0,8 MPa for non-treated wood, and 1,6 MPa for treated wood. The gluing pressure recommended for finger-joints is 9 MPa, for treated wood or not.

Foi provado por Byde que é possível adicionar um fim do tipo Delaunay a uma variedade compacta não degenerada de curvatura escalar constante positiva; desde que ela seja localmente conformemente plana em alguma vizinhança do ponto de colagem. A variedade resultante é não-compacta e possui a mesma curvatura escalar constante. O principal objetivo desta tese é generalizar este resultado. Construiremos uma família a um parâmetro de soluções para o problema de Yamabe singular positivo em qualquer variedade compacta não degenerada cujo tensor de Weyl anula-se até uma ordem suficientemente grande no ponto singular. Se a dimensão da variedade é no máximo 5; nenhuma condição sobre o tensor de Weyl é necessária. Usaremos técnicas de pertubação e o método de colagem. _________________________________________________________________________________________ ABSTRACT: It has been showed by Byde [5] that it is possible to attach a Delaunay type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this thesis is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to suciently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.

The thesis deals with the theory of 3D printing, materials for printing and bonding. It focuses on FDM printing and PLA material. It also compares gluing with other methods and deals with the distribution of adhesives, the gluing procedure and tests of glued joints. The work, based on literature studies, proposes an experiment to evaluate selected adhesives. The practical part contains a description, results and evaluation of the experiment. The evaluation of adhesives is based on tensile and shear strength.

Detta projekt har utförts på monteringslinan i Volvo Construction Equipment (VCE), Hallsberg. Där har man en robotcell som arbetar i samspel med limstationerna 17&18. Utav 4 hyttfamiljer limmas och monteras glasrutor till 3 av hyttfamiljerna manuellt. Robotcellen limmar glasrutorna för 1 hyttfamilj som har 2 typer av hytter, med en cykeltid på 8,4–9,5 min. Då takttiden ligger på 7,5 min/hytt för resten av linan anses limstationerna vara en trång sektor enligt VCE. Med en ökad framtida produktvariation och ett ökat kundbehov anser VCE att den trånga sektorn har förutsättningarna för att bli en framtida flaskhals då takttiden ska ner till 7,2 min/hytt till år 2020. Med hjälp av Leanprinciper och Sex Sigma verktyg har projektutförarna avhandlat frågeställningen "Hur kan robotcellen effektiviseras med avseende på cykeltid och framtida produktvariation?". Genom en framtagning av en nulägesanalys med hjälp av tidmätningar, intervjuer och teoriinsamling kunde sedan en värdeflödeskarta skapas. Genom rotorsaksanalyser på icke-värdeskapande men nödvändiga aktiviteter kunde ett förbättringsförslag presenteras med en uppskattad cykeltid på mindre än 7–7,6 min. Detta med hjälp av en hanteringsrobot och en limrobot som har separata aktiviteter som är parallella, istället för att ha en robot som har samtliga aktiviteter som i nuläget. Projektutförarna rekommenderar att VCE för arbetet vidare genom att- ta reda på hur lång tid det tar för en automatiserad limning av glasrutor till samtliga hyttyper, - undersöka hur snabbt hanteringsroboten behöver arbeta i förhållande till limroboten för att tillfredsställa behovet av glasrutor för limroboten, - undersöka hur kasserade glasrutor från robotcellerna påverkar monteringsflödet samt - undersöka utvecklingsmöjligheter för att effektivisera den enda värdeskapande aktiviteten för limroboten med avseende på cykeltid. / This project has been carried out on the assembly line in Volvo Construction Equipment (VCE), Hallsberg. A robotic cell is located in the assembly line and works in conjunction with the glue stations 17 & 18. Out of 4 cab families, windows are glued and mounted to 3 of the cab families manually. The robotic cell glues the windows for 1 cab family that has 2 types of cabs, with a cycle time of 8.4-9.5 min. Since the takt time is 7.5 min/cab in the other sections of the assembly line, the glue stations are considered a narrow sector according to VCE. With increased future product variability and increased customer demand, VCE considers that the narrow sector has the prerequisites for becoming a future bottleneck, as the takt time will drop to 7.2 min/cab to 2020. With the help of Lean principles and Six Sigma tools, project executives have dealt with the question "How can the robotic cell be made more efficient with respect to cycle time and future product variability?”. Through a current state analysis by means of time measurements, interviews and theory collection, a value-flow chart could then be created. By root cause analysis on non-value-adding but necessary activities an improvement proposal could be presented with an estimated cycle time of less than 7-7.6 min. The thesis workers propose that VCE goes ahead by- finding out how long an automated gluing of windows take (for all cabs),- examining how fast the handling robot needs to work in relation to the glue robot (to satisfy the need for windows for the glue robot),- examining how discarded windows from the robotic cells affect the assembly flow, and- investigating the opportunities to streamline the only value-creating activity for the glue robot with regards to cycle time.

La formulation de nouveaux adhésifs durant les dernières décennies a permis de développer des procédés de collage innovants pour l’industrie du bois. Dans cette optique, le collage du bois à l’état vert permet de créer une chaîne d’élaboration de matériaux techniques, sans ruptures de charges dans le processus industriel grâce à un séchage en fin de cycle. Or le collage implique un retrait empêché des assemblages au cours du séchage, induisant un développement de contraintes. À ce jour, de nombreuses études numériques portant sur le comportement du bois collé soumis à des variations hydriques ont été effectuées, sans pour autant avoir été mises en relation avec des études expérimentales. L’extension récente des techniques de corrélation d'images permet d'effectuer des mesures de champs sans contact, particulièrement adaptées à un matériau hygroscopique tel que le bois. L’objectif de la thèse est d’appréhender par des approches expérimentale et numérique les contraintes hydromécaniques qui peuvent survenir dans des assemblages contrecollés. Une expérimentation, permettant de mesurer les champs durant le séchage a été mise en place. Le modèle numérique retenu est un modèle élastique orthotrope, permettant de prendre en compte la morphologie du matériau bois. Nous pouvons identifier les paramètres gouvernant les mécanismes d'endommagement du joint de colle et proposer des conditions de mise en œuvre optimisant la tenue de l'assemblage au séchage. / The development of new adhesives during the last decades has implied radically changes in the sawmill industry. The gluing of unseasonned wood, called green gluing, allows a new order in the process of engineered wood products, thanks to the final position of the drying step. Nevertheless, the jointing prevents the free shrinkage of wood during the mass transfer, involving a stress development. Nowadays, numerous drying models have been developed to predict the deformations and stresses of wood, but few have been confirmed by experimental tests. For an organic material such as wood, the measurement of hygromechanical properties will be more efficient if there is no interaction between the material and the measurement system. One of the most promising experimental techniques for this kind of issue is noncontact optical measurement based on Digital Image Correlation principles. The objective of this study is to evaluate by experimental and numerical solutions the stress development in green-glued wood laminated timber subjected to drying conditions. An experimental method has been set up and the results allow to validate a numerical finite elements model. The model which has been chosen is an orthotropic elastic model, able to describe the morphology of the wood material. This study allows to identify the parameters governing the damage of the interface. This approach leads to suggest some preconizationsin order to optimize the behavior of product during the drying procedure.

L'objet de cette thèse consiste en la construction de nouveaux exemples de surfaces (ou hypersurfaces) minimales dans les espaces euclidiens R^3, R^n x R avec n>2 ou dans l'espace homogène S^2 x R. Nous prouvons l'existence de surfaces minimales dans R^3 arbitrairement proches d'un polygone convexe. Nous prouvons également l'existence d'hypersurfaces minimales de type Riemann dans R^n x R, n>2. Celles-ci peuvent être interprétées comme étant une famille d'hyperplans horizontaux (des bouts) reliés les uns aux autres par des morceaux de caténoïdes déformés (des cous). Nous donnons un résultat général pour ce type d'objet quand il est périodique ou bien quand il a un nombre fini de bouts horizontaux. Cela se fait sous certaines hypothèses de contraintes sur les forces intervenant dans la construction. Nous finissons en donnant plusieurs exemples, notamment l'existence d'une hypersurface de type Wei verticale qui n'existe pas en dimension 3. Nous donnons aussi la preuve de l'existence d'une surface minimale de type Riemann dans S^2 x R telle que deux bouts sphériques sont reliés entre eux alternativement par 1 cou et 2 cous. Là aussi, nous mettons en évidence le rôle joué par les forces lors de la construction. De même que dans le chapitre précédent, la méthode repose sur un processus de recollement. Nous donnons une description très précise de la caténoïde et la surface de Riemann dans S^2 x R. Enfin, nous établissons l'existence dans R^n x R d'hypersurfaces de type Scherk lorsque n>2 / This thesis is devoted to the construction of numerous examples of minimal surfaces (or hypersurfaces) in the $3$-Euclidean space, R^n x R with n>2 or in the homogeneous space S^2 x R . We prove the existence of minimal surfaces in R^3 as close as we want of a convex polygon. We prove the existence of minimal hypersurfaces in R^n x R, n>2, whose have Riemann's type. These ones could be considered as a family of horizontal hyperplanes (the ends) which are linked to each other by pieces of deformed catenoids (the necks). We provide a general result in the case simply-periodic together with the case of a finite number of hyperplanar ends. Our construction lies on some conditions associates with the forces that characterize the different configurations. We end with giving some examples ; in particular, we exhibit the existence of vertical Wei example that does not exists in the 3-dimensional case. We also prove the existence of the analogous of the Wei example in S^2 x R. The surface is such that two spherical ends are linked by 1 neck and 2 necks alternatively. Here again, we highlight the role that the forces play in the construction. Moreover, like in the previous chapter, the method lies on a gluing process. We give an accurate description of the catenoid and the Riemann's minimal example in S^2 x R. Finally, we demonstrate the existence of Scherk type hypersurfaces in R^n x R when n>2