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Index and nullity of proper biharmonic maps in spheres

S. Montaldo, C. Oniciuc, A. Ratto

TL;DR

This work advances the spectral analysis of the second variation of the bienergy $E_2$ for proper biharmonic maps into spheres by deriving explicit formulas for the iterated Jacobi operator $I_2$ in key equivariant settings. It decomposes the domain into Laplacian-eigenfunction blocks, yielding exact eigenvalues and multiplicities on families like $oldsymbol{ m T^2 o S^2}$ and $oldsymbol{ m S^1 o S^2}$, and provides concrete index and nullity formulas, including conjectural stability patterns. A central contribution is the introduction and computation of reduced index and nullity in equivariant contexts, together with stability results for noncompact domains (e.g., $oldsymbol{ m R o S^2}$) and extensions to Legendre immersions and conformal diffeomorphisms, plus generalizations to ellipsoids $oldsymbol{ m Q^n(b)}$. The findings deliver exact stability data that deepen understanding of bienergy landscapes and guide future geometric analysis of biharmonic maps into spheres and related symmetric spaces.

Abstract

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyse a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps $\varphi:\mathbb{R} \to \mathbb{S}^2$ is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations.

Index and nullity of proper biharmonic maps in spheres

TL;DR

This work advances the spectral analysis of the second variation of the bienergy for proper biharmonic maps into spheres by deriving explicit formulas for the iterated Jacobi operator in key equivariant settings. It decomposes the domain into Laplacian-eigenfunction blocks, yielding exact eigenvalues and multiplicities on families like and , and provides concrete index and nullity formulas, including conjectural stability patterns. A central contribution is the introduction and computation of reduced index and nullity in equivariant contexts, together with stability results for noncompact domains (e.g., ) and extensions to Legendre immersions and conformal diffeomorphisms, plus generalizations to ellipsoids . The findings deliver exact stability data that deepen understanding of bienergy landscapes and guide future geometric analysis of biharmonic maps into spheres and related symmetric spaces.

Abstract

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyse a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations.

Paper Structure

This paper contains 11 sections, 30 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Proof of the results
  3. Proof of Theorem \ref{['Spectrum-theorem-toro-sfera']}
  4. Proof of Theorem \ref{['Index-theorem-toro-sfera']}
  5. Proof of Theorem \ref{['Index-theorem-r-harmonic-circles']}
  6. Proof of Theorem \ref{['th: Sasahara_T2_S5']}
  7. Proof of Theorem \ref{['th:noncompact']}
  8. Reduced Index and Nullity
  9. Proof of Proposition \ref{['Index-theorem-equivariant']}
  10. Conformal diffeomorphisms
  11. Further developments

Key Result

Theorem 1.1

Let $\varphi:M\to N$ be a biharmonic map between two Riemannian manifolds $(M,g)$ and $(N,h)$, where $M$ is compact. Then the Hessian of the bienergy functional $E_2$ at a critical point $\varphi$ is given by where $I_2 \,:\mathcal{C}\left(\varphi^{-1} TN\right) \to \mathcal{C}\left(\varphi^{-1} TN\right)$ is a semilinear elliptic operator of order $4$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 1.9
  • Theorem 1.10: S1
  • ...and 49 more