On Closed Conformal Vector Fields with Singularities

Abstract Let $$M^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:math> be a complete, connected, noncompact m -dimensional Riemannian manifold and $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> be a nontrivial closed conformal vector field on M , with at least one singular point, say p , and conformal factor $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> . We show that, when $$m&gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> or when $$m=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and the singular set of $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> consists of isolated points at which $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> does not vanish, then p is the only singular point of $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> and $$\exp _p:T_pM\rightarrow M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>exp</mml:mo> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> is a diffeomorphism. Then, we use this fact to present a formula, built on $$|\xi |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> , for the Riemannian volume of geodesic balls of M centered at p . When $$m=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 $$\mathbb {C}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:math> is the only example in the class of Kähler manifolds of complex dimension $$m&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .