The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories.
When you zoom out it appears to be a smooth sheet, but when you zoom in, it is a bunch of dots connected by lines or loops.
There is a smooth proper family with connected smooth base manifold T, ( p : {mathcal {X}}rightarrow T) having two fibres respectively isomorphic to X, and Y, where Y is one of the 4 surfaces ( S = (C_1 times C_2) / G), ( S_ : = (overline{C_1} times C_2) / G), ( bar{S }= (overline{C_1} times overline{C_2} ) / G), ( S_ : = (C_1 times overline{C_2} ) / G = overline{ S_).
The distance in (G_{k}) is defined as follows: For any ((x,t,m), tilde {x},tilde{t},j in G_{k}), suppose (l((x,t,m), (tilde{x},tilde{t},j))) is a smooth curve in (G_{k}) that connects the points ((x,t,m)) and ((tilde{x},tilde{t},j)), and let (|l((x,t,m), (tilde{x},tilde{t},j))|) denote the length of the curve.
We study the boundary value problem of the quasi-linear elliptic equationdiv(|∇u|m−2∇u)+f x,u,∇u)= 0in Ω,u= 0on ∂Ω, where Ω⊂Rn (n⩾2) is a connected smooth domain, and the exponent m∈(1,n) is a positive number.
We now specialize to the case when (X) is a connected smooth manifold and (G) a topological group acting on (X) properly and effectively by diffeomorphisms.
