Posted online: 2018-09-04 13:28:56Z by Benoît R. Kloeckner917
Cite as: P-180904.1
Denote by $I_\kappa^n(v)$ the perimeter (i.e. $(n-1)$-dimensional volume of the boundary) of a round ball of volume $v$ in the simply connected Riemannian manifold $X_\kappa^n$ of dimension $n$ and constant curvature $\kappa$. It is known that every domain $D\subset X_\kappa^n$ has perimeter at least $I_\kappa^n(\lvert D\rvert)$, where $\lvert D\rvert$ denotes the volume of $D$.
Conjecture - Let $M$ be a simply connected Riemannian manifold $M$ of dimension $n$ with sectional curvature bounded above by $\kappa\le 0$. Every domain $D\subset M$ has perimeter at least $I_\kappa^n(\lvert D\rvert)$.
The conjecture is known to be true in dimension 2 (Weil), 3 (Kleiner), and 4 when either $\kappa=0$ (Croke) or $\lvert D\rvert\le v_0$ for some explicit $v_0=v_0(\kappa)$ (Kloeckner-Kuperberg).
In dimension 4, Kloeckner and Kuperberg reduce the conjecture to an inequality for a vectorial ODE. In the same work, a variant of the conjecture for $\kappa>0$ is proposed (simple connectedness is replaced by $D$ having at most one geodesic between each pair of points) and solved in dimensions 2 and 4 .
The conjecture is completely open in dimension 5 and above, no technique used in lower dimension being applicable.
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