By deriving Reilly type inequality we prove a lower bound estimate of the first Dirichlet eigenvalue of the Finsler Laplacian operator on a compact Finsler measure space with smooth boundary <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_445_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$(M,F,d\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and weighted Ricci curvature bound from below <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_445_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </InlineMediaObject> <EquationSource Format="TEX">$Ric _N\ge (N-1)k>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>i</mi> <msub> <mi>c</mi> <mi>N</mi> </msub> <mo>≥</mo> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, whose boundary has non-positive <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_445_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$d\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>-mean curvature. Moreover, we show that the lower bound is achieved if and only if M is isometric to a forward (backward) geodesic ball of radius <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_445_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{\pi }{2\sqrt{k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <msqrt> <mi>k</mi> </msqrt> </mrow> </mfrac> </math></EquationSource> </InlineEquation> and F has constant radial flag curvature equal to k.

First Dirichlet Eigenvalue Estimates and Rigidity for a Finsler Measure Space

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发表日期：

2025年3月3日

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Songting E-Yin,
Xiaohuan Mo

摘要

By deriving Reilly type inequality we prove a lower bound estimate of the first Dirichlet eigenvalue of the Finsler Laplacian operator on a compact Finsler measure space with smooth boundary $(M,F,d\mu )$ $(M, F, d μ)$ and weighted Ricci curvature bound from below $Ric _N\ge (N-1)k>0$ $R i c_{N} \geq (N - 1) k > 0$ , whose boundary has non-positive $d\mu $ $d μ$ -mean curvature. Moreover, we show that the lower bound is achieved if and only if M is isometric to a forward (backward) geodesic ball of radius $\frac{\pi }{2\sqrt{k}}$ $\frac{π}{2 \sqrt{k}}$ and F has constant radial flag curvature equal to k.