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<p>The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schrödinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is possible to obtain a lower bound on the fundamental gap in terms of the diameter of the domain [<CitationRef CitationID="CR22">22</CitationRef>]. In this article, we prove this conjecture by establishing conformal log-concavity estimates for the first eigenfunction. This builds off earlier work by the authors and Saha [<CitationRef CitationID="CR17">17</CitationRef>] as well as recent work by Cho, Wei and Yang [<CitationRef CitationID="CR7">7</CitationRef>]. We also prove spectral gap estimates for a more general class of problems on conformally flat manifolds and investigate the relationship between the gap and the inradius. For example, we establish gap estimates for domains in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {S}^1 \times \mathbb {S}^{N-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>1</mn> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> which are convex with respect to the universal affine cover.</p>

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Spectral Gap Estimates on Conformally Flat Manifolds

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Gabriel Khan,
  • Malik Tuerkoen

摘要

The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schrödinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is possible to obtain a lower bound on the fundamental gap in terms of the diameter of the domain [22]. In this article, we prove this conjecture by establishing conformal log-concavity estimates for the first eigenfunction. This builds off earlier work by the authors and Saha [17] as well as recent work by Cho, Wei and Yang [7]. We also prove spectral gap estimates for a more general class of problems on conformally flat manifolds and investigate the relationship between the gap and the inradius. For example, we establish gap estimates for domains in \(\mathbb {S}^1 \times \mathbb {S}^{N-1}\) S 1 × S N - 1 which are convex with respect to the universal affine cover.