A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

Vladimir Rovenski,Sergey Stepanov,Irina Tsyganok

doi:10.3390/axioms10040333

Vladimir Rovenski, Sergey Stepanov + Show 1 more

Open Access

https://doi.org/10.3390/axioms10040333

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Abstract

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.

Highlights

The prototype of the generalized Bochner technique is the celebrated classical Bochner technique, first introduced by S
We have a number of theorems based on the classical Bochner technique, which usually show that the assumption of positive or negative curvature sectional curvatures of compact Riemannian manifolds yields the vanishing of some geometrically interesting tensor fields and mappings
We discuss the global geometry of conformal mappings of complete Riemannian and Kähler manifolds using a generalized version of the Bochner technique

Summary

Introduction

The prototype of the generalized Bochner technique is the celebrated classical Bochner technique, first introduced by S. Others was later called the generalized Bochner technique (see, for example, [8]) This method studies the relationship between the geometry of a complete. We discuss the global geometry of conformal mappings of complete Riemannian and Kähler manifolds using a generalized version of the Bochner technique. This article continues the series of works [12,13] and can demonstrate to both newcomers to the field and experienced geometers various methods of the generalized Bochner technique for research on the example of conformal mappings. In the other four sections, we demonstrate applications of various methods of the generalized Bochner technique to the study of conformal diffeomorphisms of complete Riemannian manifolds

Preliminaries on Conformal Mappings and the Classical Bochner Technique

An Application of the Theory of Subharmonic Functions to the Study of

An Application of the Theory of Convex Functions to the Study of

An Application to the Study of Conformal Transformations of the Mixed

Conclusions