Constant potentials do not minimize the fundamental gap on convex domains in negatively curved Hadamard manifolds

We show that for every negatively curved Hadamard manifold $X$ and every $D>0$ there exists a convex domain $\mathrm{\Omega }\subseteq X$ with diameter $D$ and a convex potential $V$ on $\mathrm{\Omega }$ such that the fundamental gap of the operator $-\mathrm{\Delta }+V$ is strictly smaller than the fundamental gap of $-\mathrm{\Delta }$. This shows that the second part of the fundamental gap conjecture is wrong in every negatively curved manifold. This is significantly harder than in the previously known case of hyperbolic space because, due to the lack of symmetry, one has to study a true PDE, and not just an ODE.