Minimal surfaces with low genus in lens spaces

Given a Riemannian <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>RP</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> RP³ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or there exist one minimal real projective plane together with two distinct minimal 2-spheres. Our proof is based on a variant multiplicity one theorem for the Simon–Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>RP</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> RP³ contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>RP</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> RP³ by the degree method. Moreover, using the same strategy, we show that, in the lens space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L(4m,2m± 1) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> m≥ 1 , with a bumpy metric or a metric of positive Ricci curvature, there either exist <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> N(m) distinct minimal Klein bottles, or there exist one minimal Klein bottle and three distinct minimal 2-spheres, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> N(1)=4 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> N(m)=2 for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> m≥ 2 , and the first case happens under the positive Ricci assumption.