On the minimal number of closed geodesics on positively curved Finsler spheres

Abstract In this paper, we prove that, for every Finsler metric on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> S^{n} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> n\geq 4 ) with reversibility 𝜆 and flag curvature 𝐾 satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>−</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mi>λ</m:mi> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>&lt;</m:mo> <m:mi>K</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> (\frac{2n-3}{n-1})^{2}(\frac{\lambda}{\lambda+1})^{2}&lt;K\leq 1 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>λ</m:mi> <m:mo>&lt;</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:math> \lambda&lt;\frac{n-1}{n-2} , there exist at least 𝑛 prime closed geodesics on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (S^{n},F) , which solves a conjecture of Katok and Anosov for such positively curved spheres when 𝑛 is even. Furthermore, if the number of closed geodesics on such positively curved Finsler <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> S^{n} is finite, then there exist at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 2[\frac{n}{2}]-1 non-hyperbolic closed geodesics.