... that the set{Y ∈ M : τ(Y ) ≤ τ (X)}has the form (∗). Since this is true for every X ∈ M, in fact all of M musthave the form (∗).3. The fundamental theorem3.1. Theorem. For 0 < α < 1, there ... prove the Arzela-Ascoli Theorem §2.7, we state the Schauder estimates for the heat equation, and we derive the nonparamet-ric equations for mean curvature flow. First we recall the definitions of the Annals ... to problem 1 if and only if (∗) holds for all (X, V ) in the domain of β(M).Similarly, the equation for problem (2) isβ(M)(X, V ) = mX,and the equation for problem (3) isβ(M)(X, V ) = −trace...