Poincare metric

날짜 : 2023-06-19 07:27

주제 :GeometryComplexKahlerEinsteinBergman

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Abstract

본 강연에서는 푸앙카레 원반의 상수곡률 계랑을 일반차원 복소다양체 위의 켈러 계량으로 일반화하는 두 가지 방법으로써, 켈러-아인슈타인 계량과 버그만 계량을 소개한 뒤, 이 둘 사이의 연관 관계와 최신 연구동향에 대해 소개하고자 한다.

내용

Kahler Geometry

Kahler Geometry = Complex Geometry $\cap$ Riemannian Geometry Kahler Manifold = $X+J+g$ satisfying $\nabla J=0$ where $J$ is a tensor and $g$ is a Riemannian metric.

What is the canonical or best metric on a given complex mfd?

1. Riemannian Geometry aspect : Kahler - Einstein Metric
2. Complex Geometry aspect : Bergman Metric which is invariant under $\text{Aut}\left(X\right)=\left\{\phi :X\to X:\text{holomorphic}\right\}$

Poincare Metric

n=1

$X$ : compact 1-dimensional complex manifold There is a uniformation theorem $d\tilde{X}=\frac{1}{{\left(1+c\left(x^2+y^2\right)\right)}^2}\left(dx\otimes dx+dy\otimes dy\right)$ denominator $\lambda$ then $K_{\hat{X}}=-\frac{\Delta \left(\log \lambda \right)}{2\lambda }=4c$ If $C=0$, $\lambda =\frac{{\partial }^2}{\partial z\partial \bar{z}}\left({\left|z\right|}^2\right)=\frac{{\partial }^2}{\partial z\partial \bar{z}}\left(\frac{1}{C}\log \left(1+C{\left|z\right|}^2\right)\right)$ Let the $\frac{1}{C}\log \left(1+C{\left|z\right|}^2\right)$ be a potential function

Higher Dimension?

No uniformation theorem $\Delta \phi =e^{\phi }$

Kahler - Einstein Metric

Riemannian metric $g$ is hemitian

If $g\left(JX,JY\right)=g\left(X,Y\right)$ Remark of metric $g$ is hermitian $\leftrightarrow$ w\left(X,Y\right) \coloneq g \left(JX, Y\right)$

Hermintian metric $g$ is Kahler

If $dw=0$. Also denote $d{z}_j:-d{x}_j+\sqrt{-1}d{y}_j$ and g_{jk}=g^{\mathbb{C}}\left(\frac{\partial}{\partial z_{j}}, \frac{\partial}{\partial \bar{z}_{k}}\right)\newline $g=\sum g_{ik}d{z}^i\otimes \overline{d{z}^k}\leftrightarrow \sqrt{-1}{\sum }_{j,k}{g}_{j,k}d{z}^j\vee \overline{d{z}^k}$

$g$ : Kahler- Einstein

If $g$: Kahler, ${\text{Ric}}_g\left(X,X\right)=\frac{{\text{Ric}}_g\left(X,X\right)}{g\left(X,X\right)}=C$

Complex Monge - Ampere EquationNon-linearPDE

Bergman metric

Kobayashi Bergman kernel form

Conclusion

1. Berman metric $B_X,w_X$ : invariant under $\text{Aut}\left(X\right)$
2. $X$ : bounded homogeneous domain $\subset {\mathbb{C}}^n$
3. $X$: cpt $\Rightarrow {\int }_X{B}_X={\text{dim}}_{\mathbb{C}}{\Omega }_2^{n,0}\left(X\right)={\text{dim}}_{\mathbb{C}}{\mathcal{H}}_2^{n,0}\left(X\right)$

출처(참고문헌)

연결문서

Complex Geometry Geometry SOP Einstein Manifold Kahler Manifold Complex Monge - Ampare Equation