<p>In this paper, we provide some rigidity results for compact manifolds with smooth boundary by using appropriate geometric or topological assumptions and the Obata-type equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nabla ^2 f -fg =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>f</mi> <mo>-</mo> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with Robin boundary condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_{\nu } = cf$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>ν</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (note that in this case <i>f</i> has no critical points). By the same idea, we also use equations <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\nabla ^2 f +fg =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>f</mi> <mo>+</mo> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nabla ^2 f=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> to establish similar rigidity results. The proofs of our rigidity results are mainly based on the warped product structures on compact manifolds determined by the Obata-type equation (see Proposition <InternalRef RefID="FPar11">1.11</InternalRef>), and we also provide the corresponding structure on complete non-compact manifolds with compact boundary (see Proposition <InternalRef RefID="FPar13">1.13</InternalRef>). It should be pointed that we have actually provided all possible structures determined by the equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\nabla ^2 f -fg =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>f</mi> <mo>-</mo> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f_{\nu } = cf$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>ν</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> for complete manifolds with compact boundary.</p>

Some Rigidity Results Related to the Obata-Type Equation

期刊论文
发表日期：

2026年7月2日

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Yiwei Liu,
Yi-Hu Yang

摘要

In this paper, we provide some rigidity results for compact manifolds with smooth boundary by using appropriate geometric or topological assumptions and the Obata-type equation $\nabla ^2 f -fg =0$ $\nabla^{2} f - f g = 0$ with Robin boundary condition $f_{\nu } = cf$ $f_{ν} = c f$ , where $c>1$ $c > 1$ (note that in this case f has no critical points). By the same idea, we also use equations $\nabla ^2 f +fg =0$ $\nabla^{2} f + f g = 0$ and $\nabla ^2 f=0$ $\nabla^{2} f = 0$ to establish similar rigidity results. The proofs of our rigidity results are mainly based on the warped product structures on compact manifolds determined by the Obata-type equation (see Proposition 1.11), and we also provide the corresponding structure on complete non-compact manifolds with compact boundary (see Proposition 1.13). It should be pointed that we have actually provided all possible structures determined by the equation $\nabla ^2 f -fg =0$ $\nabla^{2} f - f g = 0$ with $f_{\nu } = cf$ $f_{ν} = c f$ for complete manifolds with compact boundary.