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A detailed description of the generalized Calabi type Kahler surfaces

Ewelina Mulawa

Abstract

In this paper we study QCH Kähler surfaces, i.e. 4-dimensional Riemannian manifolds (of signature (++++)) admitting a Kähler complex structure with quasi-constant holomorphic sectional curvature. We give a detailed description of QCH Kähler surfaces of generalized Calabi type.

A detailed description of the generalized Calabi type Kahler surfaces

Abstract

In this paper we study QCH Kähler surfaces, i.e. 4-dimensional Riemannian manifolds (of signature (++++)) admitting a Kähler complex structure with quasi-constant holomorphic sectional curvature. We give a detailed description of QCH Kähler surfaces of generalized Calabi type.
Paper Structure (4 sections, 8 theorems, 127 equations)

This paper contains 4 sections, 8 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Hermitian surfaces
  3. Properties of Hermitian surfaces with Hermitian Ricci tensor and Kähler natural opposite structure
  4. A description of the generalized Calabi type Kähler surfaces which are not of Calabi type

Key Result

Theorem 4.1

Let $U\subset \mathbb R^2$ and let $g_{\Sigma}=h^2(dx^2+dy^2)$ be a Riemannian metric on $U$, where $h:U \rightarrow \mathbb R$ is a positive function $h=h(x,y)$. Let $\omega_{\Sigma}=h^2dx \wedge dy$ be the volume form of $\Sigma=(U,g)$. Let $M=U \times N$, where $N=\{(z,t) \in \mathbb R^2: z < 0\} and the function $H$ satisfies the equation $\Delta \ln H=(\ln H)_{xx}+(\ln H)_{yy}=2h^2$ on $U$, $

Theorems & Definitions (14)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • proof
  • Theorem 4.6
  • proof
  • Theorem 4.7
  • proof
  • ...and 4 more