Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces

Abstract

Let $\mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{\infty }$ compact connected manifold, possibly with boundary. Fix a smoothness class $\mathcal{F}=C^{\infty }$, Hölder $C^{k,\alpha }$ or Sobolev $W^{k,p}$. The space $\mathcal{F}(V,\mathcal{G})$ resp. $\mathcal{F}(V,M)$ of all $\mathcal{F}$ maps $V\to \mathcal{G}$ resp. $V\to M$ is a Banach/Fréchet Lie group resp. complex manifold. Let ${\mathcal{F}}^0(V,\mathcal{G})$ resp. ${\mathcal{F}}^0(V,M)$ be the component of $\mathcal{F}(V,\mathcal{G})$ resp. $\mathcal{F}(V,M)$ containing the identity resp. constants. A map $f$ from a domain $\Omega \subset {\mathcal{F}}_1(V,M)$ to ${\mathcal{F}}_2(W,M)$ is called range decreasing if $f(x)(W)\subset x(V)$, $x\in \Omega$. We prove that if ${\dim }_{\mathbb{R}}\mathcal{G}\ge 2$, then any range decreasing group homomorphism $f:{\mathcal{F}}_1^0(V,\mathcal{G})\to {\mathcal{F}}_2(W,\mathcal{G})$ is the pullback by a map $\varphi :W\to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $\Omega$ $\to$ ${\mathcal{F}}_2(W,M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps ${\mathcal{F}}_1(V,N)\supset \Omega \to {\mathcal{F}}_2(W,M)$. In particular, we identify some classes of holomorphic maps ${\mathcal{F}}_1^0(V,{\mathbb{P}}^n)\to {\mathcal{F}}_2(W,{\mathbb{P}}^m)$, including all automorphisms of ${\mathcal{F}}^0(V,{\mathbb{P}}^n)$.

26 pages

References in corpus (1)

• Optimality for totally real immersions and independent mappings of manifolds into C^N