# Creating and Flattening Cusp Singularities by Deformations of Bi-conformal Energy

## TL;DR

This paper explores the creation and flattening of cusp singularities in domains via bi-conformal energy mappings, extending geometric function theory to include degenerate elliptic PDE systems.

## Contribution

It introduces a bi-conformal variant of the Riemann Mapping Theorem, characterizing boundary singularities achievable through bi-conformal energy mappings.

## Key findings

- Characterization of boundary singularities created by bi-conformal energy mappings
- Extension of quasiconformal homeomorphisms to degenerate elliptic PDEs
- Identification of domains with non-quasiball singular boundaries

## Abstract

Mappingsofbi-conformalenergyformthewidestclass of homeomorphisms that one can hope to build a viable extension of Geometric Function Theory with connections to mathematical models of Nonlinear Elasticity. Such mappings are exactly the ones with finite conformal energy and integrable inner distortion. It is in this way, that our studies extend the applications of quasiconformal homeomorphisms to the degenerate elliptic systems of PDEs. The present paper searches a bi-conformal variant of the Riemann Mapping Theorem, focusing on domains with exemplary singular boundaries that are not quasiballs. We establish the sharp description of boundary singularities that can be created and flattened by mappings of bi-conformal energy.