In this paper, we study geometric structures of compact static vacuum spaces and function theoretic properties for the potential functions satisfying the static vacuum equation. In particular, we investigate geometric conditions under which static vacuum spaces are warped product and Bach-flat. As an application, we prove that if a triple <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(M^n, g, f), n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mi>n</mi> </msup> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, is a compact static vacuum space satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\omega :=df \wedge i_{\nabla f}{\mathring{\textrm{Ric}}} = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>:</mo> <mo>=</mo> <mi>d</mi> <mi>f</mi> <mo>∧</mo> <msub> <mi>i</mi> <mrow> <mi mathvariant="normal">∇</mi> <mi>f</mi> </mrow> </msub> <mover accent="true"> <mtext>Ric</mtext> <mo>˚</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, then M is either isometric to a round sphere or a warped product of a circle with a compact Einstein manifold of positive Ricci curvature, up to finite cover. Furthermore, if (M, g) has positive isotropic curvature, then M is either isometric to a round sphere or a product <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {S}}^1 \times {\mathbb {S}}^{n-1}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>1</mn> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

On the Structure of Compact Static Vacuum Spaces and Positive Isotropic Curvature

期刊论文
发表日期：

2026年3月11日

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Seungsu Hwang,
Gabjin Yun

摘要

In this paper, we study geometric structures of compact static vacuum spaces and function theoretic properties for the potential functions satisfying the static vacuum equation. In particular, we investigate geometric conditions under which static vacuum spaces are warped product and Bach-flat. As an application, we prove that if a triple $(M^n, g, f), n \ge 4$ $(M^{n}, g, f), n \geq 4$ , is a compact static vacuum space satisfying $\omega :=df \wedge i_{\nabla f}{\mathring{\textrm{Ric}}} = 0$ $ω : = d f \land i_{\nabla f} \overset{˚}{Ric} = 0$ , then M is either isometric to a round sphere or a warped product of a circle with a compact Einstein manifold of positive Ricci curvature, up to finite cover. Furthermore, if (M, g) has positive isotropic curvature, then M is either isometric to a round sphere or a product ${\mathbb {S}}^1 \times {\mathbb {S}}^{n-1}.$ $S^{1} \times S^{n - 1} .$