Extremal Beltrami Differentials of Non-Landslide Type

Abstract

In this paper it is shown that there are infinitely many extremal Beltrami differentials of non-landside type and non-constant modulus in a Teichmüller equivalence class if the class contains a landslide extremal. The result answers, affirmatively in a stronger sense, a problem posed by Li.

Nondecreasable and Weakly Nondecreasable Dilatations

“…Here, we generalize the definition for general Beltrami differentials. It was proved by Fan [2] and the author [16] independently that if µ contains more than one extremal, then it contains an infinite number of extremals of nonlandslide type.…”

Infinitesimally extremal Beltrami differentials of non-landslide type

“…In [2], Fan answered the problem affirmatively and proved that if [µ] contains infinitely many extremals, then there always exist infinitely many extremals of nonlandslide type in [µ]. The author gave a more precise formulation for the problem in [13] by use of variability set and point shift differentials, that is, Theorem B. Let [µ] be given in T (S).…”

“…The author gave a more precise formulation for the problem in [13] by use of variability set and point shift differentials, that is, Theorem B. Let [µ] be given in T (S).…”

Non-decreasable extremal Beltrami differentials of non-landslide type

“…The conception of non-landslide was firstly introduced by Li in [8]. It was proved by Fan [3] and the author [22] independently that if µ contains more than one extremal, then it contains infinitely many extremals of non-landslide type.…”