# On the index of minimal 2-tori in the 4-sphere

## TL;DR

This paper establishes lower bounds on the Morse index of minimal 2-tori in spheres, characterizing cases of equality and extending results to tori with vanishing Hopf differential.

## Contribution

It proves the minimal 2-torus in S^4 has index at least 6, with equality only for the Clifford torus, and generalizes index bounds for tori with vanishing Hopf differential in higher spheres.

## Key findings

- Minimal 2-tori in S^4 have Morse index ≥ 6.
- Equality in index occurs only for the Clifford torus.
- Tori with vanishing Hopf differential in S^n have index ≥ n+3.

## Abstract

In this note we prove that any minimal $2$-torus in $S^4$ has Morse index at least $6$, with equality if and only if it is congruent to the Clifford torus in some great $S^3\subset S^4$.For a minimal $2$-torus in $S^n$ with vanishing Hopf differential, we show that its index is at least $n+3$, and that this estimate is sharp: the equilateral $2$-torus fully embedded in $S^5\subset S^n$ as a homogeneous minimal surface in $S^n$ has index exactly $n+3$.