Index one minimal surfaces in positively curved $3$-manifolds

TL;DR

This paper constructs a specific Riemannian metric on the 3-dimensional projective space that contains a minimal surface of genus 3 with index 1, illustrating a new example in positively curved 3-manifolds.

Contribution

It provides the first example of a minimal surface with genus 3 and index 1 in a positively curved 3-manifold, expanding understanding of minimal surface configurations.

Findings

Constructed a positive curvature metric on $ ext{RP}^3$

Found a minimal surface of genus 3 with index 1

Demonstrated existence of such surfaces in positively curved spaces

Abstract

We construct a Riemannian metric of positive sectional curvature on the $3$-dimensional projective space with a two-sided closed embedded minimal surface of genus $3$, index $1$ and nullity $0$.