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

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 $0$.

Figures (7)

• Figure 1:
• Figure 2: Picture of the Euclidean orbifold $T^3(1)/I222$ appearing in the PhD thesis of Dunbar dunbar, p 51. The underlying space is the projective space given as an Euclidean $3$-ball with antipodal points identified. The $1$-cycle inside the ball represents the singular set of the projective manifold.
• Figure 3: The fundamental piece $S$ of Schwarz' $P$ minimal surface in $C(1/2)$ has genus $0$ and is orthogonal to the boundary of the cube along six congruent convex curves. So, it is a free boundary minimal surface in the cube and identifying the opposite faces we obtain the Schwarz P minimal surface in the 3-torus $\Sigma\subset T^3(1/2)$.
• Figure 4: The Square Catenoid is a minimal annulus bounded by two parallel squares of side $\sqrt{2}/4$ which differ by the vertical translation $(0,0,1/4)$. It decomposes into twelve congruent minimal quadrilaterals and the Schwarz' $P$ surface is the union of two square catenoids.
• Figure 5: The fundamental domain of the surface $\Sigma^{-1}$ on the cube of side $1$ is formed by eight copies of the fundamental piece $S$ of the Schwarz surface $\Sigma$.

Theorems & Definitions (9)

• Theorem 1
• Remark 1
• Theorem 2: Ross ross
• Corollary 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof