Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> be a complete, connected, noncompact m-dimensional Riemannian manifold and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> be a nontrivial closed conformal vector field on M, with at least one singular point, say p, and conformal factor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>. We show that, when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> or when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and the singular set of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> consists of isolated points at which <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> does not vanish, then p is the only singular point of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\exp _p:T_pM\rightarrow M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>exp</mo> <mi>p</mi> </msub> <mo>:</mo> <msub> <mi>T</mi> <mi>p</mi> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> is a diffeomorphism. Then, we use this fact to present a formula, built on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$|\xi |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>ξ</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, for the Riemannian volume of geodesic balls of M centered at p. When <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, such a formula generates necessary and sufficient conditions for M to be: (i) conformally equivalent to the Euclidean or hyperbolic plane; (ii) of finite total curvature. Finally, after showing that the conformal factor can be prescribed under some conditions, we finish the paper proving that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {C}^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> is the only example in the class of Kähler manifolds of complex dimension <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

On Closed Conformal Vector Fields with Singularities

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2026年2月6日

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Antonio Caminha,
F. Yure Jansel

摘要

Let $M^m$ $M^{m}$ be a complete, connected, noncompact m-dimensional Riemannian manifold and $\xi $ $ξ$ be a nontrivial closed conformal vector field on M, with at least one singular point, say p, and conformal factor $\psi $ $ψ$ . We show that, when $m>2$ $m > 2$ or when $m=2$ $m = 2$ and the singular set of $\xi $ $ξ$ consists of isolated points at which $\psi $ $ψ$ does not vanish, then p is the only singular point of $\xi $ $ξ$ and $\exp _p:T_pM\rightarrow M$ ${exp}_{p} : T_{p} M \to M$ is a diffeomorphism. Then, we use this fact to present a formula, built on $|\xi |$ $| ξ |$ , for the Riemannian volume of geodesic balls of M centered at p. When $m=2$ $m = 2$ , such a formula generates necessary and sufficient conditions for M to be: (i) conformally equivalent to the Euclidean or hyperbolic plane; (ii) of finite total curvature. Finally, after showing that the conformal factor can be prescribed under some conditions, we finish the paper proving that $\mathbb {C}^m$ $C^{m}$ is the only example in the class of Kähler manifolds of complex dimension $m>1$ $m > 1$ .