Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symphonic map from ellipsoid to ellipsoid

Xiangzhi Cao

TL;DR

The paper establishes the existence of Symphonic maps from ellipsoid to ellipsoid and provides a Hopf construction of such maps. It develops a variational framework via a functional J(φ) on ellipsoid joins, derives the associated Euler–Lagrange equations, and proves existence and regularity of minimizers, thereby yielding symphonic joins between ellipsoids. Additionally, it adapts the Hopf construction to the ellipsoidal context, deriving ODE conditions for the angular function φ that ensure symphonicity and proving the existence of Hopf-derived symphonic maps, including the case of Hopf maps to ellipsoids. This work extends the theory of symphonic maps to ellipsoidal domains and targets, offering new tools for representing elements in homotopy groups via symphonic mappings.

Abstract

In this paper, we proved the existence of Symphonic map from ellipsoid to ellipsoid. We also geive give Hopf construction of Symphonic map from ellipsoid to ellipsoid.

Symphonic map from ellipsoid to ellipsoid

TL;DR

The paper establishes the existence of Symphonic maps from ellipsoid to ellipsoid and provides a Hopf construction of such maps. It develops a variational framework via a functional J(φ) on ellipsoid joins, derives the associated Euler–Lagrange equations, and proves existence and regularity of minimizers, thereby yielding symphonic joins between ellipsoids. Additionally, it adapts the Hopf construction to the ellipsoidal context, deriving ODE conditions for the angular function φ that ensure symphonicity and proving the existence of Hopf-derived symphonic maps, including the case of Hopf maps to ellipsoids. This work extends the theory of symphonic maps to ellipsoidal domains and targets, offering new tools for representing elements in homotopy groups via symphonic mappings.

Abstract

In this paper, we proved the existence of Symphonic map from ellipsoid to ellipsoid. We also geive give Hopf construction of Symphonic map from ellipsoid to ellipsoid.
Paper Structure (3 sections, 23 theorems, 84 equations)

This paper contains 3 sections, 23 theorems, 84 equations.

Table of Contents

  1. Backgroud
  2. Preliminary Knowledge
  3. Hopf construction of symphonic map

Key Result

Lemma 2.1

Let $\varphi$ be an harmonic map from Riemannian manifold $(M, g)$ into ellipsoid $Q^n(c, d).$ Let $\Phi=i \circ \varphi$ be the composition of $\varphi$ with the canonical embedding $i$ of $Q^n(c, d)$ into $\mathbb{R}^{n+1}$. Then $\Phi$ is harmonic iff where If we write $\Phi=\left(\Phi_1, \Phi_2\right)$, the components being the projections on the factors of the ambient space $\mathbb{R}^{

Theorems & Definitions (44)

  • Lemma 2.1: cf. MR1044657MR1022743
  • Lemma 2.2
  • Remark 2.1
  • proof
  • Lemma 2.3
  • Lemma 2.4: cf. Lemma 1 in MR3238295
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Remark 2.2
  • ...and 34 more