Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> be a free boundary biharmonic hypersurface in the Euclidean unit ball <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {B}^{m+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Denote by H the mean curvature function on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> satisfies a sharp linear isoperimetric inequality <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m\textrm{Vol}(\Sigma ) \le \textrm{Vol}(\partial \Sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mtext>Vol</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mtext>Vol</mtext> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where equality holds if and only if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is a free boundary minimal hypersurface. Moreover, we prove that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is minimal if either H is constant along the boundary or <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H\frac{\partial H}{\partial \nu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mfrac> <mrow> <mi>∂</mi> <mi>H</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is nonpositive along the boundary, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> denotes the outward unit conormal vector. These results can be thought of as a partial affirmative answer to Chen’s conjecture.

Rigidity of Free Boundary Biharmonic Hypersurfaces in the Unit Ball

期刊论文
发表日期：

2026年3月12日

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Keomkyo Seo,
Gabjin Yun

摘要

Let $\Sigma $ $Σ$ be a free boundary biharmonic hypersurface in the Euclidean unit ball $\mathbb {B}^{m+1}$ $B^{m + 1}$ . Denote by H the mean curvature function on $\Sigma $ $Σ$ . We prove that $\Sigma $ $Σ$ satisfies a sharp linear isoperimetric inequality $m\textrm{Vol}(\Sigma ) \le \textrm{Vol}(\partial \Sigma )$ $m Vol (Σ) \leq Vol (\partial Σ)$ , where equality holds if and only if $\Sigma $ $Σ$ is a free boundary minimal hypersurface. Moreover, we prove that $\Sigma $ $Σ$ is minimal if either H is constant along the boundary or $H\frac{\partial H}{\partial \nu }$ $H \frac{\partial H}{\partial ν}$ is nonpositive along the boundary, where $\nu $ $ν$ denotes the outward unit conormal vector. These results can be thought of as a partial affirmative answer to Chen’s conjecture.