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This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.
Book Synopsis The Defect Relation of Meromorphic Maps on Parabolic Manifolds by : George Lawrence Ashline
Download or read book The Defect Relation of Meromorphic Maps on Parabolic Manifolds written by George Lawrence Ashline and published by American Mathematical Soc.. This book was released on 1999 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.
Value distribution theory studies the behavior of mermorphic maps. Let f: M - N be a merom orphic map between complex manifolds. A target family CI ~ (Ea1aEA of analytic subsets Ea of N is given where A is a connected. compact complex manifold. The behavior of the inverse 1 family ["'(CI) = (f- {E )laEA is investigated. A substantial theory has been a created by many contributors. Usually the targets Ea stay fixed. However we can consider a finite set IJ of meromorphic maps g : M - A and study the incidence f{z) E Eg(z) for z E M and some g E IJ. Here we investigate this situation: M is a parabolic manifold of dimension m and N = lP n is the n-dimensional projective space. The family of hyperplanes in lP n is the target family parameterized by the dual projective space lP* We obtain a Nevanlinna theory consisting of several n First Main Theorems. Second Main Theorems and Defect Relations and extend recent work by B. Shiffman and by S. Mori. We use the Ahlfors-Weyl theory modified by the curvature method of Cowen and Griffiths. The Introduction consists of two parts. In Part A. we sketch the theory for fixed targets to provide background for those who are familar with complex analysis but are not acquainted with value distribution theory.
Book Synopsis Value Distribution Theory for Meromorphic Maps by : Wilhelm Stoll
Download or read book Value Distribution Theory for Meromorphic Maps written by Wilhelm Stoll and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: Value distribution theory studies the behavior of mermorphic maps. Let f: M - N be a merom orphic map between complex manifolds. A target family CI ~ (Ea1aEA of analytic subsets Ea of N is given where A is a connected. compact complex manifold. The behavior of the inverse 1 family ["'(CI) = (f- {E )laEA is investigated. A substantial theory has been a created by many contributors. Usually the targets Ea stay fixed. However we can consider a finite set IJ of meromorphic maps g : M - A and study the incidence f{z) E Eg(z) for z E M and some g E IJ. Here we investigate this situation: M is a parabolic manifold of dimension m and N = lP n is the n-dimensional projective space. The family of hyperplanes in lP n is the target family parameterized by the dual projective space lP* We obtain a Nevanlinna theory consisting of several n First Main Theorems. Second Main Theorems and Defect Relations and extend recent work by B. Shiffman and by S. Mori. We use the Ahlfors-Weyl theory modified by the curvature method of Cowen and Griffiths. The Introduction consists of two parts. In Part A. we sketch the theory for fixed targets to provide background for those who are familar with complex analysis but are not acquainted with value distribution theory.
Few people outside of mathematics are aware of the varieties of mathemat ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function.
Book Synopsis Geometry V by : Robert Osserman
Download or read book Geometry V written by Robert Osserman and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: Few people outside of mathematics are aware of the varieties of mathemat ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function.
This book is intended for graduate students and research mathematicians working in partial differential equations.
Book Synopsis Realization of Vector Fields and Dynamics of Spatially Homogeneous Parabolic Equations by : Edward Norman Dancer
Download or read book Realization of Vector Fields and Dynamics of Spatially Homogeneous Parabolic Equations written by Edward Norman Dancer and published by American Mathematical Soc.. This book was released on 1999 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians working in partial differential equations.
This memoir considers the Dirichlet problem for parabolic operators in a half space with singular drift terms. Chapter I begins the study of a parabolic PDE modelled on the pullback of the heat equation in certain time varying domains considered by Lewis-Murray and Hofmann-Lewis. Chapter II obtains mutual absolute continuity of parabolic measure and Lebesgue measure on the boundary of this halfspace and also that the $L DEGREESq(R DEGREESn)$ Dirichlet problem for these PDEs has a solution when $q$ is large enough. Chapter III proves an analogue of a theorem of Fefferman, Kenig, and Pipher for certain parabolic PDEs with singular drift terms. Each of the chapters that comprise this memoir has its own numbering system and list
Book Synopsis The Dirichlet Problem for Parabolic Operators with Singular Drift Terms by : Steve Hofmann
Download or read book The Dirichlet Problem for Parabolic Operators with Singular Drift Terms written by Steve Hofmann and published by American Mathematical Soc.. This book was released on 2001 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir considers the Dirichlet problem for parabolic operators in a half space with singular drift terms. Chapter I begins the study of a parabolic PDE modelled on the pullback of the heat equation in certain time varying domains considered by Lewis-Murray and Hofmann-Lewis. Chapter II obtains mutual absolute continuity of parabolic measure and Lebesgue measure on the boundary of this halfspace and also that the $L DEGREESq(R DEGREESn)$ Dirichlet problem for these PDEs has a solution when $q$ is large enough. Chapter III proves an analogue of a theorem of Fefferman, Kenig, and Pipher for certain parabolic PDEs with singular drift terms. Each of the chapters that comprise this memoir has its own numbering system and list
Let $V = {\mathbb R}^{p,q}$ be the pseudo-Euclidean vector space of signature $(p,q)$, $p\ge 3$ and $W$ a module over the even Clifford algebra $C\! \ell^0 (V)$. A homogeneous quaternionic manifold $(M,Q)$ is constructed for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \wedge^2 W \rightarrow V$. If the skew symmetric vector valued bilinear form $\Pi$ is nondegenerate then $(M,Q)$ is endowed with a canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is a homogeneous quaternionic pseudo-Kahler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kahler manifold $(M,Q,g)$ is shown to admit a simply transitive solvable group of automorphisms. In this special case ($p=3$) we recover all the known homogeneous quaternionic Kahler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If $p>3$ then $M$ does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kahler manifolds. Then it is shown that for $q = 0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed with a Riemannian metric $h$ such that $(M,Q,h)$ is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if $p>3$. The twistor bundle $Z \rightarrow M$ and the canonical ${\mathrm SO}(3)$-principal bundle $S \rightarrow M$ associated to the quaternionic manifold $(M,Q)$ are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution $\mathcal D$ of complex codimension one, which is a complex contact structure if and only if $\Pi$ is nondegenerate. Moreover, an equivariant open holomorphic immersion $Z \rightarrow \bar{Z}$ into a homogeneous complex manifold $\bar{Z}$ of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \vee^2 W \rightarrow V$ a homogeneous quaternionic supermanifold $(M,Q)$ is constructed and, moreover, a homogeneous quaternionic pseudo-Kahler supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form $\Pi$ is nondegenerate.
Book Synopsis A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures by : Vicente Cortés
Download or read book A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures written by Vicente Cortés and published by American Mathematical Soc.. This book was released on 2000 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $V = {\mathbb R}^{p,q}$ be the pseudo-Euclidean vector space of signature $(p,q)$, $p\ge 3$ and $W$ a module over the even Clifford algebra $C\! \ell^0 (V)$. A homogeneous quaternionic manifold $(M,Q)$ is constructed for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \wedge^2 W \rightarrow V$. If the skew symmetric vector valued bilinear form $\Pi$ is nondegenerate then $(M,Q)$ is endowed with a canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is a homogeneous quaternionic pseudo-Kahler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kahler manifold $(M,Q,g)$ is shown to admit a simply transitive solvable group of automorphisms. In this special case ($p=3$) we recover all the known homogeneous quaternionic Kahler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If $p>3$ then $M$ does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kahler manifolds. Then it is shown that for $q = 0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed with a Riemannian metric $h$ such that $(M,Q,h)$ is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if $p>3$. The twistor bundle $Z \rightarrow M$ and the canonical ${\mathrm SO}(3)$-principal bundle $S \rightarrow M$ associated to the quaternionic manifold $(M,Q)$ are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution $\mathcal D$ of complex codimension one, which is a complex contact structure if and only if $\Pi$ is nondegenerate. Moreover, an equivariant open holomorphic immersion $Z \rightarrow \bar{Z}$ into a homogeneous complex manifold $\bar{Z}$ of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \vee^2 W \rightarrow V$ a homogeneous quaternionic supermanifold $(M,Q)$ is constructed and, moreover, a homogeneous quaternionic pseudo-Kahler supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form $\Pi$ is nondegenerate.
This book is intended for graduate students and research mathematicians interested in global analysis, analysis on manifolds, and symplectic geometry.
Book Synopsis Periodic Hamiltonian Flows on Four Dimensional Manifolds by : Yael Karshon
Download or read book Periodic Hamiltonian Flows on Four Dimensional Manifolds written by Yael Karshon and published by American Mathematical Soc.. This book was released on 1999 with total page 87 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians interested in global analysis, analysis on manifolds, and symplectic geometry.
We develop the theory of compactness of maps between toposes, together with associated notions of separatedness. This theory is built around two versions of "propriety" for topos maps, introduced here in a parallel fashion. The first, giving what we simply call "proper" maps, is a relatively weak condition due to Johnstone. The second kind of proper maps, here called "tidy", satisfy a stronger condition due to Tierney and Lindgren. Various forms of the Beck-Chevalley condition for (lax) fibered product squares of toposes play a central role in the development of the theory. Applications include a version of the Reeb stability theorem for toposes, a characterization of hyperconnected Hausdorff toposes as classifying toposes of compact groups, and of strongly Hausdorff coherent toposes as classifiying toposes of profinite groupoids. Our results also enable us to develop further particular aspects of the factorization theory of geometric morphisms studied by Johnstone. Our final application is a (so-called lax) descent theorem for tidy maps between toposes. This theorem implies the lax descent theorem for coherent toposes, conjectured by Makkai and proved earlier by Zawadowski.
Book Synopsis Proper Maps of Toposes by : Ieke Moerdijk
Download or read book Proper Maps of Toposes written by Ieke Moerdijk and published by American Mathematical Soc.. This book was released on 2000 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop the theory of compactness of maps between toposes, together with associated notions of separatedness. This theory is built around two versions of "propriety" for topos maps, introduced here in a parallel fashion. The first, giving what we simply call "proper" maps, is a relatively weak condition due to Johnstone. The second kind of proper maps, here called "tidy", satisfy a stronger condition due to Tierney and Lindgren. Various forms of the Beck-Chevalley condition for (lax) fibered product squares of toposes play a central role in the development of the theory. Applications include a version of the Reeb stability theorem for toposes, a characterization of hyperconnected Hausdorff toposes as classifying toposes of compact groups, and of strongly Hausdorff coherent toposes as classifiying toposes of profinite groupoids. Our results also enable us to develop further particular aspects of the factorization theory of geometric morphisms studied by Johnstone. Our final application is a (so-called lax) descent theorem for tidy maps between toposes. This theorem implies the lax descent theorem for coherent toposes, conjectured by Makkai and proved earlier by Zawadowski.
This book describes the theories and developments in Nevanlinna theory and Diophantine approximation. Although these two subjects belong to the different areas: one in complex analysis and one in number theory, it has been discovered that a number of striking similarities exist between these two subjects. A growing understanding of these connections has led to significant advances in both fields. Outstanding conjectures from decades ago are being solved.Over the past 20 years since the first edition appeared, there have been many new and significant developments. The new edition greatly expands the materials. In addition, three new chapters were added. In particular, the theory of algebraic curves, as well as the algebraic hyperbolicity, which provided the motivation for the Nevanlinna theory.
Book Synopsis Nevanlinna Theory And Its Relation To Diophantine Approximation (Second Edition) by : Min Ru
Download or read book Nevanlinna Theory And Its Relation To Diophantine Approximation (Second Edition) written by Min Ru and published by World Scientific. This book was released on 2021-03-10 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the theories and developments in Nevanlinna theory and Diophantine approximation. Although these two subjects belong to the different areas: one in complex analysis and one in number theory, it has been discovered that a number of striking similarities exist between these two subjects. A growing understanding of these connections has led to significant advances in both fields. Outstanding conjectures from decades ago are being solved.Over the past 20 years since the first edition appeared, there have been many new and significant developments. The new edition greatly expands the materials. In addition, three new chapters were added. In particular, the theory of algebraic curves, as well as the algebraic hyperbolicity, which provided the motivation for the Nevanlinna theory.
This volume contains the Proceedings of the International Workshop "Complex Analysis", which was held from February 12-16, 1990, in Wuppertal (Germany) in honour of H. Grauert, one of the most creative mathematicians in Complex Analysis of this century. In complete accordance with the width of the work of Grauert the book contains research notes and longer articles of many important mathematicians from all areas of Complex Analysis (Altogether there a re 49 articles in the volume). Some of the main subjects are: Cau chy-Riemann Equations with estimates, q-convexity, CR structures, deformation theory, envelopes of holomorphy, function algebras, complex group actions, Hodge theory, instantons, Kähler geometry, Lefschetz theorems, holomorphic mappings, Nevanlinna theory, com plex singularities, twistor theory, uniformization.
Book Synopsis Complex Analysis by : Klas Diederich
Download or read book Complex Analysis written by Klas Diederich and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the Proceedings of the International Workshop "Complex Analysis", which was held from February 12-16, 1990, in Wuppertal (Germany) in honour of H. Grauert, one of the most creative mathematicians in Complex Analysis of this century. In complete accordance with the width of the work of Grauert the book contains research notes and longer articles of many important mathematicians from all areas of Complex Analysis (Altogether there a re 49 articles in the volume). Some of the main subjects are: Cau chy-Riemann Equations with estimates, q-convexity, CR structures, deformation theory, envelopes of holomorphy, function algebras, complex group actions, Hodge theory, instantons, Kähler geometry, Lefschetz theorems, holomorphic mappings, Nevanlinna theory, com plex singularities, twistor theory, uniformization.