Existence of extremal Beltrami coefficients with non-constant modulus.pdf

a r X i v : 0 7 0 5 .1 6 5 5 v 1 [ m a t h .C V ] 1 1 M a y 2 0 0 7 Existence of extremal Beltrami coefficients with non-constant modulus GUOWU YAO ABSTRACT. Suppose [μ]T (?) is a point of the universal Teichmu?ller space T (?). In 1998, it was shown by Boz?in et al. that there exists μ such that μ has non-constant modulus and is uniquely extremal in [μ]T (?). It is a natural problem whether there is always an extremal Beltrmai coefficient of constant modulus in [μ]T (?) if [μ]T (?) admits more than one extremal Beltrami coefficient. The purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered and an open problem is proposed. 1 . Introduction Suppose D is a Jordan domain in the complex plane C and w = f(z) be a quasi- conformal mapping on D. The complex dilatation of f is defined by μ(z) = fz?(z) fz(z) , which is also called the Beltrami coefficient of f . Let M(D) be the open unit ball of L∞(D). Let z1, z2, z3 be three boundary points on ?D. For a given μ ∈ M(D), denote by fμ the uniquely determined quasiconformal mapping of D onto itself with complex dilatation μ and normalized to fix z1, z2, z3. The elements of M(D) are also called Beltrami coefficients. Two elements μ and ν in M(D) are Teichmu?ller equivalent, which is denoted by μ ～ ν, if fμ|?D = f ν|?D. Then T (D) = M(D)/ ～ is the Teichmu?ller space of D. The equivalence class of the Beltrami coefficient zero is the basepoint of T (D). Given μ ∈ M(D), we denote by [μ]T (D) the set of all elements ν ∈ M(D) equivalent to μ, and set (1. 1) k(μ) = inf{‖ν‖∞ : ν ∈ [μ]}. 2000 Mathematics Subject Classification. Primary 30C75; Secondary 30C62. Key words and phrases. Teichmu?ller space, Delta Inequality, Beltrami coefficient, extremal set. The research was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science