Scientific report on the second summer institute, several complex variables. Part II. Complex manifolds

Lectures on Riemannian Geometry, Part II: Complex Manifolds

This is a set of introductory lecture notes on the geometry of complex manifolds. It is the second part of the course on Riemannian Geometry given at the MRI Masterclass in Mathematics, Utrecht, 2008. The first part was given by Prof. E. van den Ban, and his lectures notes can be found on the web-site of this course, http://www.math.uu.nl/people/ban/riemgeom2008/riemgeom2008.html. Topics that w...

Lecture notes on several complex variables

Complex Dynamics in Several Variables

1. Motivation 117 2. Iteration of Maps 118 3. Regular Versus Chaotic Behavior 119 4. The Horseshoe Map and Symbolic Dynamics 120 5. Hénon Maps 123 6. Properties of Horseshoe and Hénon Maps 126 7. Dynamically Defined Measures 127 8. Potential Theory 129 9. Potential Theory in One-Variable Dynamics 131 10. Potential Theory and Dynamics in Two Variables 133 11. Currents and Applications to Dynamic...

Polynomially bounded solutions of the Loewner‎ ‎differential equation in several complex variables

‎We determine the‎ ‎form of polynomially bounded solutions to the Loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form $f(z,t)=e^{int_0^t A(tau){rm d}tau}z+cdots$‎, ‎where‎ ‎$A:[0,infty]rightarrow L(mathbb{C}^n,mathbb{C}^n)$ is a locally‎ ‎Lebesgue integrable mapping and satisfying the condition‎ ‎$$sup_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[A(tau)...

A short introduction to several complex variables

This text is a short two hour introduction into the theory of several complex variables. These lectures were given in May 1996 at Caltech to the class Ma 108 substituting for someone else. As prerequisite, the topic requires some familiarity with complex analysis in one dimension. The higher dimensional generalization of complex analysis in one variables is an important part of mathematics. Our...