Dissertations / Theses on the topic 'Complex Monge Ampere'
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Zhang, Xiangwen. "Complex Monge-Ampere equation and its applications in complex geometry." Thesis, McGill University, 2012. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=107800.
Full textDans cette thèse, il est question de l'étude des équations de type Monge-Ampère complexes. On y présente une analyse basée sur les différentes techniques utilisées dans la théorie des équations aux dérivées partielles ainsi que certaines applications géométriques. En premier lieu, nous présentons l'estimation à priori des équations de type Hessienne complexes sur des variétés hermitiennes. Ces estimations sont indispensables à la résolution de ces équations par le biais des méthodes de continuité. Au fait, nous établirons des estimations sur la première et la seconde dérivée des équations Monge-Ampère complexes de la même manière faite par Yau sur les variétés kählériennes.Au troisième chapitre, nous étudions la régularité de Hölder intérieure des dérivées secondes de la solution pour les équations de type Monge-Ampère complexes. De plus, en visant la généralisation de ce type de résultats de régularité à des géométries plus généralee, on a obtenu une estimation de deuxième ordre de type Bedford-Taylor classique et une version locale des estimations de Calabi de troisième ordre sur des variétés hermitiennes. Les deux derniers chapitres de cette thèse sont consacrés aux problèmes géométriques reliés aux équations de type Monge-Ampère complexes. Nous donnons quelques résultats sur la représentation non négative pour la classe de frontière du cône de Kähler et l'existence des métriques généralisée Kähler-Einstein.
Guidi, Chiara. "The Complex Monge-Ampère Equation." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9004/.
Full textDo, Hoang Son. "Equations de Monge-Ampère complexes paraboliques." Thesis, Toulouse 3, 2015. http://www.theses.fr/2015TOU30113/document.
Full textThe aim of this thesis is to make a contribution to understanding parabolic complex Monge-Ampère equations on domains of Cn. Our study is centered around cases where the initial condition is irregular. We want to prove the existence of solutions which satisfies continuity up to the boundary and continuity up to the initial time
Tô, Tat Dat. "Flots de Monge-Ampère complexes sur les variétés hermitiennes compactes." Thesis, Toulouse 3, 2018. http://www.theses.fr/2018TOU30072/document.
Full textIn this thesis we study the complex Monge-Ampère flows, and their generalizations and geometric applications on compact Hermitian manifods. In the first two chapters, we prove that a general complex Monge-Ampère flow on a compact Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti- Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. This part gave rise to two independent publications. In the third chapter, a notion of parabolic C -subsolution is introduced for parabolic non-linear equations, extending the theory of C -subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows. This part is a joint work with Duong H. Phong (Columbia University) In the fourth chapter, a viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in Cn. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions. This part is a joint work with Sl-awomir Dinew (Jagiellonian University) and Hoang-Son Do (Hanoi Institute of Mathematics)
Ivarsson, Björn. "Regularity and boundary behavior of solutions to complex Monge–Ampère equations." Doctoral thesis, Uppsala University, Department of Mathematics, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-1603.
Full textIn the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.
Ivarsson, Björn. "Regularity and boundary behaviour of solutions to complex Monge-Ampère equations /." Uppsala : Matematiska institutionen, Univ. [distributör], 2002. http://publications.uu.se/theses/91-506-1533-5/.
Full textSun, Wei. "On a Class of Complex Monge-Ampère Type Equations on Hermitian Manifolds." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1366286119.
Full textDi, Nezza Eleonora. "Géométrie des équations de Monge-Ampère complexes sur des variétés kähleriennes compactes." Toulouse 3, 2014. http://thesesups.ups-tlse.fr/2351/.
Full textIn the mid 70's, Aubin-Yau solved the problem of the existence of Kaehler metrics with constant negative or identically zero Ricci curvature on compact Kaehler manifolds. In particular, they proved the existence and regularity of the solution of the complex Monge-Ampère equation (*) (\omega+dd^c\f)^n=f\omega^n dollar dollar where dollar \mu dollar is a positive measure, it is not always possible to make sense of the left-hand side of (*). It was nevertheless observed by Guedj and Zeriahi thata construction going back to Bedford and Taylor enables in this global setting to define the non-pluripolar part of the would-be positive measure dollar (\theta+dd^c\f)^n dollar for an arbitrary dollar \thetadollar-psh function, where dollar\theta dollar represents a big class. The notion of big classes is invariant by bimeromorphism while this is not the case in the Kaehler setting. It is therefore natural to study the invariance property of the non-pluripolar product in the wider context of big cohomology classes. We indeed show that it is a bimeromorphic invariant. Generalizing the Aubin-Mabuchi energy functional, Boucksom, Eyssidieux, Guedj et Zeriahi introduced weighted energies associated to big cohomology classes. Under some natural assumptions, we show that such energies are also bimeromorphic invariants. We also investigate probability measures with finite energy and we show that this notion is a biholomorphic but not a bimeromorphic invariant. Furtheremore, we give criteria insuring that a given measure has finite energy and test these on various examples. We then study complex Monge-Ampère equations on quasi-projective varieties. In particular we consider a compact Kaehler manifold dollar X dollar, dollar D\subset X dollar a divisor and we look at the equation dollar dollar(\omega+dd^c\f)^n=f\omega^n dollar dollar where dollar f dollar is smooth outside dollar D dollar and with a precise behavior near the divisor. We prove that the unique normalized solution dollar \f dollar is smooth outside dollar D dollar and we are able to describe its asymptotic behavior near dollar D dollar (joint work with Hoang Chinh Lu). The solution is clearly not bounded in general and thus the idea is to find a convenient ``model" function (a priori singular) bounding from below the solution. To do so we introduce generalized Monge-Ampère capacities, and use them following Kolodziej's approach who deals with globally bounded potentials. These capacities, which generalize the Bedford-Taylor Monge-Ampère capacity, turn out to be the key point when investigating the existence and the regularity of solutions of complex Monge-Ampère equations of type dollar dollar \MA(\f)=e^{\lambda\f}f \omega^n, \qquad \lambda\in \R dollar dollar where dollar f dollar has divisorial singularities. We also treat some cases when dollar f dollar is not in L^1, an important issue for the existence of singular Kaehler-Einstein metrics on general type varieties with log-canonical singularities
Charabati, Mohamad. "Le problème de Dirichlet pour les équations de Monge-Ampère complexes." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30001/document.
Full textIn this thesis we study the regularity of solutions to the Dirichlet problem for complex Monge-Ampère equations and also for complex Hessian equations in a bounded domain of Cn. In the first chapter, we give basic facts in pluripotential theory. In the second chapter, we study the modulus of continuity of solutions to the Dirichlet problem for complex Monge-Ampère equations when the right hand side is a measure with continuous density with respect to the Lebesgue measure in a bounded strongly hyperconvex Lipschitz domain. In the third chapter, we prove the Hölder continuity of solutions to this problem for some general measures. In the fourth chapter, we consider the Dirichlet problem for complex Hessian equations when the right hand side depends on the unknown function. We give a sharp estimate of the modulus of continuity of the solution as the density is continuous. Moreover, for the case of Lp-density we demonstrate that the solution is Hölder continuous up to the boundary
Babaee, Ghasemabadi Farhad. "Complex tropical currents." Thesis, Bordeaux, 2014. http://www.theses.fr/2014BORD0071/document.
Full textTo a tropical p-cycle VT in Rn, we naturally assoicate a closed (p, p)-dimensional current of order zero on (C)n denoted bu T p n(VT). Such e "tropical current" T p n(VT) cannot be an integration current along any analytic set since its support has the form log -1(VT) (C)n, where log is the coordinate-wise valuation with log(I.I). We provide sufficient (local) conditions on a tropical p-cycle such that its associated tropical is "strongly extremal" in Dop, p((C)n). In particular, if these conditions hokd for the effective cycles, then the associated current are extremal in the cone of strongly positive closed currents of bidimension (p, p) on (C)n. Nexte we explain how to extend the currents and extremality results to CPn. Further, we demonstrate how to use the intersection theory of currents to derive an intersection theory for the inderlying tropical cycles. The explicit calculations will be established by using e formula for the real Monge-Ampère measure of a tropical polynomial. Moreoer, since such tropical currents are obtained by an averaging of integration currents on toric sets, an equality between toric intersection multipmicities and the tropical multiplicities is readily settled. Finally, we explain certain relations between approximation problems of tropical cycles by amoebas of algebraic cycles and approximations of the associated currents by positive multiples of integration currents along analytic cycles. Il will be discussed haw these approximtion problems are related to a stronger formulation of the celebrated hodge conjecture
Auvray, Hugues. "Équation de Monge-Ampère complexe, métriques kählériennes de type Poincaré et instantons gravitationnels ALF." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2012. http://tel.archives-ouvertes.fr/tel-00750891.
Full textZhang, Lizhi. "The Painlevé property and nonintegrability; The Dirichlet Boundary Value Problem for Complex Monge-Ampére Type Equation." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1313540635.
Full textHed, Lisa. "Approximation and Subextension of Negative Plurisubharmonic Functions." Licentiate thesis, Umeå : Department of Mathematics and Mathematical Statistics, Umeå University, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-1799.
Full textBahraini, Alireza. "Supersymétrie et géometrie complexe." Paris 7, 2004. http://www.theses.fr/2004PA077007.
Full textPhạm, Hoàng Hiệp. "Dirichlet's problem in Pluripotential Theory." Doctoral thesis, Umeå universitet, Matematik och matematisk statistik, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-1562.
Full textSjöström, Dyrefelt Zakarias. "K-stabilité et variétés kähleriennes avec classe transcendante." Thesis, Toulouse 3, 2017. http://www.theses.fr/2017TOU30126/document.
Full textIn this thesis we are interested in questions of geometric stability for constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class. As a starting point we develop generalized notions of K-stability, extending a classical picture for polarized manifolds due to G. Tian, S. Donaldson, and others, to the setting of arbitrary compact Kähler manifolds. We refer to these notions as cohomological K-stability. By contrast to the classical theory, this formalism allows us to treat stability questions for non-projective compact Kähler manifolds as well as projective manifolds endowed with non-rational polarizations. As a first main result and a fundamental tool in this thesis, we study subgeodesic rays associated to test configurations in our generalized sense, and establish formulas for the asymptotic slope of a certain family of energy functionals along these rays. This is related to the Deligne pairing construction in algebraic geometry, and covers many of the classical energy functionals in Kähler geometry (including Aubin's J-functional and the Mabuchi K-energy functional). In particular, this yields a natural potential-theoretic aproach to energy functional asymptotics in the theory of K-stability. Building on this foundation we establish a number of stability results for cscK manifolds: First, we show that cscK manifolds are K-semistable in our generalized sense, extending a result due to S. Donaldson in the projective setting. Assuming that the automorphism group is discrete we further show that K-stability is a necessary condition for existence of constant scalar curvature Kähler metrics on compact Kähler manifolds. More precisely, we prove that coercivity of the Mabuchi functional implies uniform K-stability, generalizing results of T. Mabuchi, J. Stoppa, R. Berman, R. Dervan as well as S. Boucksom, T. Hisamoto and M. Jonsson for polarized manifolds. This gives a new and more general proof of one direction of the Yau-Tian-Donaldson conjecture in this setting. The other direction (sufficiency of K-stability) is considered to be one of the most important open problems in Kähler geometry. We finally give some partial results in the case of compact Kähler manifolds admitting non-trivial holomorphic vector fields, discuss some further perspectives and applications of the theory of K-stability for compact Kähler manifolds with transcendental cohomology class, and ask some questions related to stability loci in the Kähler cone
Kemppe, Berit. "Boundary values of plurisubharmonic functions and related topics." Doctoral thesis, Umeå universitet, Matematik och matematisk statistik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-29870.
Full textHed, Lisa. "The plurisubharmonic Mergelyan property." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-52229.
Full textDelcroix, Thibaut. "Métriques de Kähler-Einstein sur les compactifications de groupes." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAM046/document.
Full textThe main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-Einstein metrics on these varieties are developed in the first part of the work, including a computation of the complex Hessian of a $Ktimes K$-invariant function on the complexification of a compact group $K$. Another step is to associate to any non-negatively curved invariant hermitian metric on an ample linearized line bundle on a group compactification a convex function with prescribed asymptotic behavior. This is used a first time to derive a formula for the alpha invariantof an ample line bundle on a Fano group compactification. This formula is obtained through the computation of the log canonical thresholds of any non-negatively curved invariant hermitian metric, and gives the sameresult, for toric manifolds, as the one we obtained before, in an article that is included in this thesis as an appendix.Then we prove the main result by obtaining $C^0$ estimates along the continuity method, using the tools developed to reduce to a real Monge-Ampère equation on a cone. The condition obtained is that the barycenter of the polytope associated to the group compactification, with respect to the Duistermaat-Heckman measure, lies in a certain zone in the polytope. This condition can be checked on examples, gives new examples of Fano Kähler-Einstein manifolds, and also gives an example that admits no Kähler-Ricci solitons. We also compute the greatest Ricci lower bound when there are no Kähler-Einstein metrics
Duong, Quang Hai. "Limites d'idéaux de fonctions holomorphes et de fonctions de Green pluricomplexes." Toulouse 3, 2013. http://thesesups.ups-tlse.fr/2011/.
Full textThe aim of this thesis is to study the convergence of pluricomplex Green functions on a bounded hyperconvex domain in Cn, with a domaine ouvert connexe and the convergence of some families of ideals in the space ouvert connexe of all holomorphic functions on ouvert connexe. The zero variety of each those ideals, consisting of all common zeros of the holomorphic functions in the ideal, is a finite set. In the first chapter, we introduce some basic notions of potential theory in several complex variables and the pluricomplex Green function with simple logarithmic poles at finitely many points. Then, we study the convergence of these functions with simple logarithmic poles at finitely many points as the poles tend to a single point. In the second chapter, we give a method to reduce the verification of the convergence of a family of ideals of holomorphic functions. More precisely, we prove two necessary and sufficient conditions for the convergence of the family of ideals. The third chapter is devoted to the study of the convergence of pluricomplex Green functions based on three distinct poles in the particular case where all the poles tend to the origin along the same asymptotic direction. We begin by studying the limit of the family of ideals of holomorphic functions based on the same points, then we give some estimates for the upper and the lower limit of the pluricomplex Green functions. Finally, using the notion of powers of ideals of holomorphic functions, we introduce a method of Rashkovskii - Thomas and study the limit of the pluricomplex Green functions in this case. In the fourth chapter, we study the convergence of pluricomplex Green functions with simple logarithmic poles at four points as the poles tend to C² for the generic case. Then we study the limit of the family of ideals of holomorphic functions based on the same points for the degenerate case. Finally, using the notion of powers of ideals of holomorphic functions and the method of Rashkovskii - Thomas, we give some estimates for the limit of the pluricomplex Green functions for a particular case
Wang, Yu. "Local Regularity of the Complex Monge-Ampere Equation." Thesis, 2013. https://doi.org/10.7916/D8NS124V.
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