<p>We generalize Kobayashi’s connected-sum inequality to the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-Yamabe invariants. As an application, we calculate the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-Yamabe invariants of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\#m_1{\mathbb {R}}{\mathbb {P}}^n\# m_2({\mathbb {R}}{\mathbb {P}}^{n-1}\times S^1)\#lH^n\#kS_+^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mi mathvariant="double-struck">R</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> <mo>#</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>#</mo> <mi>l</mi> <msup> <mi>H</mi> <mi>n</mi> </msup> <mo>#</mo> <mi>k</mi> <msubsup> <mi>S</mi> <mo>+</mo> <mi>n</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, provided <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k+l\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>+</mo> <mi>l</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. As a corollary, we prove that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\mathbb {R}}{\mathbb {P}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> minus finitely many disjoint <i>n</i>-balls have the same <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results [<CitationRef CitationID="CR3">3</CitationRef>] on the Yamabe invariants of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\mathbb {R}}{\mathbb {P}}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Yamabe invariant of certain compact manifolds with boundary

期刊论文
发表日期：

2025年12月4日

查看全文

Xuan Yao

摘要

We generalize Kobayashi’s connected-sum inequality to the $\lambda $ $λ$ -Yamabe invariants. As an application, we calculate the $\lambda $ $λ$ -Yamabe invariants of $\#m_1{\mathbb {R}}{\mathbb {P}}^n\# m_2({\mathbb {R}}{\mathbb {P}}^{n-1}\times S^1)\#lH^n\#kS_+^n$ $# m_{1} R P^{n} # m_{2} (R P^{n - 1} \times S^{1}) # l H^{n} # k S_{+}^{n}$ , for any $\lambda \in [0,1]$ $λ \in [0, 1]$ , $n\ge 3$ $n \geq 3$ , provided $k+l\ge 1$ $k + l \geq 1$ . As a corollary, we prove that ${\mathbb {R}}{\mathbb {P}}^n$ $R P^{n}$ minus finitely many disjoint n-balls have the same $\lambda $ $λ$ -Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results [3] on the Yamabe invariants of ${\mathbb {R}}{\mathbb {P}}^3$ $R P^{3}$ .