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New constructions of biharmonic polynomial maps between spheres

Rares Ambrosie

TL;DR

The paper advances the construction and understanding of proper biharmonic maps between spheres by analyzing diagonal and product-type maps built from homogeneous polynomial components. It derives key bitension-field expressions and develops new construction techniques: a degree-elevating scheme from a harmonic degree-$k$ form $G$ to a form $F$ yielding proper biharmonicity on $S^m$ only when $m=1$, and a product-diagonal approach that yields proper biharmonic maps when radii are balanced at $r_1=r_2=1/ ext{sqrt}(2)$ and energy densities differ. It corroborates the theory with explicit examples, including Veronese-type maps and mixed-degree constructions, thereby broadening the catalog of known proper biharmonic maps between spheres and linking to prior results on harmonic homogeneous polynomial maps. The results provide systematic tools for constructing higher-dimensional proper biharmonic maps and illuminate the role of energy density and degree in these maps' harmonicity and biharmonicity properties.

Abstract

In this paper, we study diagonal maps between spheres given by two homogeneous polynomial maps between spheres, defined on the same domain sphere. First we find their bitension field, then we give a method for generating proper biharmonic maps between spheres, relying on harmonic homogeneous polynomial maps of different degrees. Further, we establish a result for constructing proper biharmonic product maps using harmonic homogeneous polynomial maps between spheres.

New constructions of biharmonic polynomial maps between spheres

TL;DR

The paper advances the construction and understanding of proper biharmonic maps between spheres by analyzing diagonal and product-type maps built from homogeneous polynomial components. It derives key bitension-field expressions and develops new construction techniques: a degree-elevating scheme from a harmonic degree- form to a form yielding proper biharmonicity on only when , and a product-diagonal approach that yields proper biharmonic maps when radii are balanced at and energy densities differ. It corroborates the theory with explicit examples, including Veronese-type maps and mixed-degree constructions, thereby broadening the catalog of known proper biharmonic maps between spheres and linking to prior results on harmonic homogeneous polynomial maps. The results provide systematic tools for constructing higher-dimensional proper biharmonic maps and illuminate the role of energy density and degree in these maps' harmonicity and biharmonicity properties.

Abstract

In this paper, we study diagonal maps between spheres given by two homogeneous polynomial maps between spheres, defined on the same domain sphere. First we find their bitension field, then we give a method for generating proper biharmonic maps between spheres, relying on harmonic homogeneous polynomial maps of different degrees. Further, we establish a result for constructing proper biharmonic product maps using harmonic homogeneous polynomial maps between spheres.

Paper Structure

This paper contains 3 sections, 21 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. New construction method of proper biharmonic maps

Key Result

Theorem 2.1

Let M be a compact manifold and consider $\psi: M \rightarrow \hbox{${\mathbb S}$}^n(r/\sqrt{2})$ a nonconstant map, where $\hbox{${\mathbb S}$}^n(r/\sqrt{2})$ is a small hypersphere of radius $r/\sqrt{2}$ of $\hbox{${\mathbb S}$}^{n+1}\left (r\right)$. The map $\phi = \hbox{${\textnormal{i}}$}\circ

Theorems & Definitions (46)

  • Theorem 2.1: LO07
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 36 more