Academic literature on the topic 'Complex Monge Ampere'
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Journal articles on the topic "Complex Monge Ampere"
Semmes, Stephen. "Complex Monge-Ampere and Symplectic Manifolds." American Journal of Mathematics 114, no. 3 (June 1992): 495. http://dx.doi.org/10.2307/2374768.
Full textDo, Hoang son. "Weak solution of Parabolic complex Monge-Ampere equation." Indiana University Mathematics Journal 66, no. 6 (2017): 1949–79. http://dx.doi.org/10.1512/iumj.2017.66.6186.
Full textChen, Xiuxiong, and Gang Tian. "Partial regularity for homogeneous complex Monge–Ampere equations." Comptes Rendus Mathematique 340, no. 5 (March 2005): 337–40. http://dx.doi.org/10.1016/j.crma.2004.11.024.
Full textMoriyón, Roberto. "The degenerate complex monge-ampere equation on thin annuli." Communications in Partial Differential Equations 11, no. 11 (January 1986): 1205–42. http://dx.doi.org/10.1080/03605308608820461.
Full textPhong, D. H., and Jacob Sturm. "The Dirichlet problem for degenerate complex Monge–Ampere equations." Communications in Analysis and Geometry 18, no. 1 (2010): 145–70. http://dx.doi.org/10.4310/cag.2010.v18.n1.a6.
Full textKolodziej, Slawomir. "The range of the complex Monge-Ampere operator II." Indiana University Mathematics Journal 44, no. 3 (1995): 0. http://dx.doi.org/10.1512/iumj.1995.44.2007.
Full textBlocki, Zbigniew. "The domain of definition of the complex Monge-Ampere operator." American Journal of Mathematics 128, no. 2 (2006): 519–30. http://dx.doi.org/10.1353/ajm.2006.0010.
Full textDinew, Slawomir, Xi Zhang, and XiangWen Zhang. "The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation." Indiana University Mathematics Journal 60, no. 5 (2011): 1713–22. http://dx.doi.org/10.1512/iumj.2011.60.4444.
Full textKoeodziej, Slawomir. "Regularity of the Entire Solutions to the Complex Monge-Ampere Equation." Communications in Analysis and Geometry 12, no. 5 (2004): 1173–84. http://dx.doi.org/10.4310/cag.2004.v12.n5.a9.
Full textÅhag, Per, Urban Cegrell, and Rafal Czyz. "On Dirichlet's principle and problem." MATHEMATICA SCANDINAVICA 110, no. 2 (June 1, 2012): 235. http://dx.doi.org/10.7146/math.scand.a-15206.
Full textDissertations / Theses on the topic "Complex Monge Ampere"
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
Books on the topic "Complex Monge Ampere"
Kołodziej, Sławomir. The complex Monge-Ampère equation and pluripotential theory. Providence, R.I: American Mathematical Society, 2005.
Find full textComplex Monge-Ampère equations and geodesics in the space of Kähler metrics. Berlin: Springer Verlag, 2012.
Find full textGuedj, Vincent, ed. Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-23669-3.
Full textTian, Gang. Complex Monge-Ampere Equation and Application on Kähler Geometry. Springer, 2006.
Find full textGuedj, Vincent. Complex Monge-Ampère Equations and Geodesics in the Space of Kähler Metrics. Springer, 2012.
Find full textBook chapters on the topic "Complex Monge Ampere"
Guan, Pengfei. "Remarks on the Homogeneous Complex Monge-Ampère Equation." In Complex Analysis, 175–85. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0009-5_10.
Full textDi Nezza, Eleonora. "The Monge-Ampère Energy Class E $$E$$." In Complex and Symplectic Geometry, 51–59. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-62914-8_4.
Full textCegrell, Urban. "Further Properties of the Monge-Ampère Operator." In Capacities in Complex Analysis, 56–65. Wiesbaden: Vieweg+Teubner Verlag, 1988. http://dx.doi.org/10.1007/978-3-663-14203-4_7.
Full textMolzon, R. "Integral Geometry of the Monge-Ampère Operator." In Contributions to Several Complex Variables, 217–26. Wiesbaden: Vieweg+Teubner Verlag, 1986. http://dx.doi.org/10.1007/978-3-663-06816-7_10.
Full textDemailly, Jean-Pierre. "Monge-Ampère Operators, Lelong Numbers and Intersection Theory." In Complex Analysis and Geometry, 115–93. Boston, MA: Springer US, 1993. http://dx.doi.org/10.1007/978-1-4757-9771-8_4.
Full textCegrell, Urban. "Plurisubharmonic Functions in ℂn — The Monge-Ampère Capacity." In Capacities in Complex Analysis, 32–55. Wiesbaden: Vieweg+Teubner Verlag, 1988. http://dx.doi.org/10.1007/978-3-663-14203-4_6.
Full textWong, Pit-Mann. "Complex Monge-Ampère equation and related problems." In Lecture Notes in Mathematics, 303–30. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0078253.
Full textBłocki, Zbigniew. "The Complex Monge–Ampère Equation in Kähler Geometry." In Lecture Notes in Mathematics, 95–141. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-36421-1_2.
Full textBoucksom, Sébastien. "Monge–Ampère Equations on Complex Manifolds with Boundary." In Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics, 257–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-23669-3_7.
Full textMaldonado, Diego. "On the Preservation of Eccentricities of Monge–Ampère Sections." In Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1), 201–31. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30961-3_12.
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