The path $$W[0,t]$$ W [ 0 , t ] of a Brownian motion on a $$d$$ d -dimensional torus $$\mathbb T ^d$$ T d run for time $$t$$ t is a random compact subset of $$\mathbb T ^d$$ T d . We study the geometric properties of the complement $$\mathbb T ^d{{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] as $$t\rightarrow \infty $$ t → ∞ for $$d\ge 3$$ d ≥ 3 . In particular, we show that the largest regions in $$\mathbb T ^d{{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] have a linear scale $$\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}$$ φ d ( t ) = [ ( d log t ) / ( d - 2 ) κ d t ] 1 / ( d - 2 ) , where $$\kappa _d$$ κ d is the capacity of the unit ball. More specifically, we identify the sets $$E$$ E for which $$\mathbb T ^d{{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] contains a translate of $$\varphi _d(t)E$$ φ d ( t ) E , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of $$\mathbb T ^d{{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] as $$t\rightarrow \infty $$ t → ∞ and the $$\varepsilon $$ ε -cover time of $$\mathbb T ^d$$ T d as $$\varepsilon \downarrow 0$$ ε ↓ 0 . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in $$\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]$$ T d \ W ρ ( t ) [ 0 , t ] , where $$W_{\rho (t)}[0,t]$$ W ρ ( t ) [ 0 , t ] is the Wiener sausage of radius $$\rho (t)$$ ρ ( t ) , with $$\rho (t)$$ ρ ( t ) chosen much smaller than $$\varphi _d(t)$$ φ d ( t ) but not too small. The idea behind this choice is that $$\mathbb T ^d {{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] consists of “lakes”, whose linear size is of order $$\varphi _d(t)$$ φ d ( t ) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of $$\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]$$ T d \ W ρ ( t ) [ 0 , t ] as $$t\rightarrow \infty $$ t → ∞ . Our results give a complete picture of the extremal geometry of $$\mathbb T ^d{{\setminus }} W[0,t]$$ T d \ W [ 0 , t ] and of the optimal strategy for $$W[0,t]$$ W [ 0 , t ] to realise extreme events.