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Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class

Egor Morozov

TL;DR

The paper constructs explicit $\mathbb S^1$-equivariant harmonic maps $u_{a,b}^{p,q,r}: (\mathbb T^2,[g_{a,b}])\to \mathbb S^3$ that generate rotationally symmetric, conformal metrics $g_{a,b}^{p,q,r}$ with controlled normalized Laplace eigenvalues. It proves the existence of a rotationally symmetric critical metric $g_{a,b}^{1,1,0}$ in non-rhombic conformal classes with $\bar{\lambda}_1(\mathbb T^2,g_{a,b}^{1,1,0})>\max\{\frac{4\pi^2}{b},8\pi\}$, refines prior results, and shows that maximal rectangular-class metrics must coincide (up to scale) with this rotational metric. The work also connects the constructed family to Otsuki tori and provides an explicit elliptic-integral parametrization; the dependence of $\bar{\lambda}_1$ on $(a,b)$ is analyzed via the Hopf differential, and a detailed discussion of limit cases and Jacobi fields illuminates the challenges in proving global maximality. Overall, the paper advances explicit construction of critical metrics in torus conformal classes and clarifies the role of symmetry and elliptic-integral structures in spectral optimization.

Abstract

We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\barλ_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\barλ_1$ in this conformal class with the value of $\barλ_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\barλ_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals.

Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class

TL;DR

The paper constructs explicit -equivariant harmonic maps that generate rotationally symmetric, conformal metrics with controlled normalized Laplace eigenvalues. It proves the existence of a rotationally symmetric critical metric in non-rhombic conformal classes with , refines prior results, and shows that maximal rectangular-class metrics must coincide (up to scale) with this rotational metric. The work also connects the constructed family to Otsuki tori and provides an explicit elliptic-integral parametrization; the dependence of on is analyzed via the Hopf differential, and a detailed discussion of limit cases and Jacobi fields illuminates the challenges in proving global maximality. Overall, the paper advances explicit construction of critical metrics in torus conformal classes and clarifies the role of symmetry and elliptic-integral structures in spectral optimization.

Abstract

We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the -th normalized Laplace-Beltrami eigenvalue functional in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain -equivariant harmonic maps . For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for in this conformal class with the value of being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for in its conformal class, then it is -equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals.
Paper Structure (15 sections, 17 theorems, 160 equations, 1 figure)

This paper contains 15 sections, 17 theorems, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Critical metrics for the normalized Laplace-Beltrami eigenvalues on surfaces
  3. The main technical results
  4. Organization of the paper
  5. Existence of τ₁, τ₂, τ₃
  6. S¹-equivariant harmonic tori in S³
  7. The spectral index of u_{a,b}{p,q,r}
  8. The value of the functional
  9. Maximal metrics in rectangular conformal classes
  10. Relation to Otsuki tori
  11. The dependence of λ̄₁ on a, b
  12. The third limit case and a nonintegrable Jacobi field
  13. Discussion
  14. Why Conjecture \ref{['con:intro-maxmetric']} is difficult?
  15. Directions for the future research

Key Result

Theorem 1

For any $(a,b)\in\mathcal{M}$ such that $a^2+b^2>1$ there exists a rotationally symmetric critical metric $g_{a,b}^{1,1,0}=\rho(y)g_{a,b}$ for $\bar{\lambda}_1$ in the conformal class $[g_{a,b}]$ such that

Figures (1)

  • Figure 1: Examples of the "top view" of the orbit space $\mathbb S_{\geqslant 0}^2$ together with the curve $\gamma_u$. Left: the nonlimit case $a=0.25,b=2.1,p=2,q=3,r=0;$Middle: the first limit case $a=0.25,b=1.25,p=1,q=2,r=0;$Right: the second limit case $a=0.5,b=2,p=2,q=3,r=1.$

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Conjecture 1.1
  • Theorem 3
  • Theorem 4: ElSI2008
  • Proposition 1.1
  • Theorem 5
  • Theorem 6
  • proof : Proof of Theorem \ref{['th:intro-harm']}
  • ...and 24 more