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• W2964996915 abstract "We show that for an<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding=application/x-tex>n</mml:annotation></mml:semantics></mml:math></inline-formula>dimensional complete non-Ricci flat gradient steady Ricci soliton with potential function<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>bounded above by a constant and curvature tensor<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R m><mml:semantics><mml:mrow><mml:mi>R</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>Rm</mml:annotation></mml:semantics></mml:math></inline-formula>satisfying<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=limit Overscript bar Endscripts Underscript r right-arrow normal infinity Endscripts r StartAbsoluteValue upper R m EndAbsoluteValue greater-than one fifth><mml:semantics><mml:mrow><mml:msub><mml:mover><mml:mo form=prefix>lim</mml:mo><mml:mo accent=false>¯<!-- ¯ --></mml:mo></mml:mover><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>r</mml:mi><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mi>R</mml:mi><mml:mi>m</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mrow><mml:annotation encoding=application/x-tex>overline {lim }_{rto infty } r|Rm|>frac {1}{5}</mml:annotation></mml:semantics></mml:math></inline-formula>,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue upper R m EndAbsoluteValue less-than-or-equal-to upper C e Superscript negative r><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mi>R</mml:mi><mml:mi>m</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mo>≤<!-- ≤ --></mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo>−<!-- − --></mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>|Rm|leq Ce^{-r}</mml:annotation></mml:semantics></mml:math></inline-formula>for some constant<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C greater-than 0><mml:semantics><mml:mrow><mml:mi>C</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>C>0</mml:annotation></mml:semantics></mml:math></inline-formula>, improving on a result of Munteanu, Sung, and Wang’s. For any four dimensional complete non-Ricci flat gradient steady Ricci soliton with scalar curvature<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S right-arrow 0><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>Sto 0</mml:annotation></mml:semantics></mml:math></inline-formula>as<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r right-arrow normal infinity><mml:semantics><mml:mrow><mml:mi>r</mml:mi><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>rto infty</mml:annotation></mml:semantics></mml:math></inline-formula>, we prove that<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue upper R m EndAbsoluteValue less-than-or-equal-to c upper S><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mi>R</mml:mi><mml:mi>m</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mo>≤<!-- ≤ --></mml:mo><mml:mi>c</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>|Rm|leq cS</mml:annotation></mml:semantics></mml:math></inline-formula>for some constant<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c greater-than 0><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>c>0</mml:annotation></mml:semantics></mml:math></inline-formula>, improving on an estimate by Cao and Cui. As an application, we show that for a four dimensional complete non-Ricci flat gradient steady Ricci soliton,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue upper R m EndAbsoluteValue><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mi>R</mml:mi><mml:mi>m</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding=application/x-tex>|Rm|</mml:annotation></mml:semantics></mml:math></inline-formula>decays exponentially provided that<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=limit Overscript bar Endscripts Underscript r right-arrow normal infinity Endscripts r upper S><mml:semantics><mml:mrow><mml:msub><mml:mover><mml:mo form=prefix>lim</mml:mo><mml:mo accent=false>¯<!-- ¯ --></mml:mo></mml:mover><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>r</mml:mi><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>overline {lim }_{rto infty } rS</mml:annotation></mml:semantics></mml:math></inline-formula>is sufficiently small and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>is bounded above by a constant." @default.
• W2964996915 created "2019-08-13" @default.
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• W2964996915 date "2019-09-25" @default.
• W2964996915 modified "2023-09-23" @default.
• W2964996915 title "Curvature estimates for steady Ricci solitons" @default.
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