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Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces

Mikhail Karpukhin, Romain Petrides, Daniel Stern

TL;DR

The paper resolves the global existence of metrics maximizing the normalized first Laplace eigenvalue on all closed surfaces by proving strict monotonicity of the conformal supremum under both cross-cap and handle attachments. It introduces refined extension operators that distinguish even/odd symmetries on the Möbius band and cylinder to obtain first-order estimates at attachment points, forcing vanishing derivatives of sphere-valued eigenmaps and leading to contradictions when maximizing would fail. Consequently, it proves $\Lambda_1(M\#\mathbb{RP}^2) > \Lambda_1(M)$ and $\Lambda_1(M\#\mathbb{T}^2) > \Lambda_1(M)$ for every closed $M$, and deduces the existence (smooth away from finitely many conical points) of $\bar{\lambda}_1$-maximizing metrics on all topologies. The results bridge orientable and nonorientable cases and connect extremal metrics to minimal immersion theory, providing a unified framework for this class of spectral optimization problems.

Abstract

Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for $λ_1$-maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum $Λ_1(M)$ of the normalized first eigenvalue over all metrics on $M$ obeys the strict monotonicity $Λ_1(M\#\mathbb{RP}^2)>Λ_1(M)$ and $Λ_1(M\#\mathbb{T}^2)>Λ_1(M)$ under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24].

Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces

TL;DR

The paper resolves the global existence of metrics maximizing the normalized first Laplace eigenvalue on all closed surfaces by proving strict monotonicity of the conformal supremum under both cross-cap and handle attachments. It introduces refined extension operators that distinguish even/odd symmetries on the Möbius band and cylinder to obtain first-order estimates at attachment points, forcing vanishing derivatives of sphere-valued eigenmaps and leading to contradictions when maximizing would fail. Consequently, it proves and for every closed , and deduces the existence (smooth away from finitely many conical points) of -maximizing metrics on all topologies. The results bridge orientable and nonorientable cases and connect extremal metrics to minimal immersion theory, providing a unified framework for this class of spectral optimization problems.

Abstract

Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for -maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum of the normalized first eigenvalue over all metrics on obeys the strict monotonicity and under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24].
Paper Structure (9 sections, 13 theorems, 156 equations)

This paper contains 9 sections, 13 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Conformal classes on connected sums from singular gluing constructions
  3. Attaching cross-caps
  4. A refined extension operator for cross-caps
  5. Proof of Proposition \ref{['cap.prop']}
  6. Attaching handles
  7. A refined extension operator for handles
  8. Proof of Proposition \ref{['hand.prop']}
  9. Proof of Theorem \ref{['gap.thm']}

Key Result

Theorem 1.1

Suppose that for any closed surface $M_0$ such that $M\approx M_0\# \mathbb{RP}^2$ or $M\approx M_0\#\mathbb{T}^2$. Then $M$ admits a $\bar{\lambda}_1$-maximizing metric, which is smooth up to finitely many isolated conical singularities.

Theorems & Definitions (24)

  • Theorem 1.1: Petrides Petrides1, Matthiesen-Siffert MS2
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2: cf. Proposition 3.1 in KNPS
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4: Lemma 8.11 in KKMS
  • ...and 14 more