In this paper, we proved that for a bounded Hopf-symmetric domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> in a non-compact rank one symmetric space M, the second Dirichlet eigenvalue <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda _2(\Omega )\le \lambda _2(B_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is a geodesic ball in M such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda _1(\Omega ) = \lambda _1(B_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This generalizes the work of Ashbaugh & Benguria, Benguria & Linde for bounded domains in constant curvature spaces.

On the PPW Conjecture for Hopf-Symmetric Sets in Non-compact Rank One Symmetric Space

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2026年1月30日

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Yusen Xia

摘要

In this paper, we proved that for a bounded Hopf-symmetric domain $\Omega $ $Ω$ in a non-compact rank one symmetric space M, the second Dirichlet eigenvalue $\lambda _2(\Omega )\le \lambda _2(B_1)$ $λ_{2} (Ω) \leq λ_{2} (B_{1})$ where $B_1$ $B_{1}$ is a geodesic ball in M such that $\lambda _1(\Omega ) = \lambda _1(B_1)$ $λ_{1} (Ω) = λ_{1} (B_{1})$ . This generalizes the work of Ashbaugh & Benguria, Benguria & Linde for bounded domains in constant curvature spaces.