Geodesics and Approximate Heat Kernels

We use the gradient flow on the path space to obtain estimates on the heat kernel k(t, x, y) on a complete Riemannian manifold. This approach gives a sharp formula for the small-time asymptotics of k(t, x, y) and an upper bound for all time for pairs x and y are not conjugate. It also gives a theorem about convolutions of heat kernels that makes precise the intuition that heat flows in packets along geodesics. On a complete Riemannian manifold M , the heat kernel k(t, x, y) of the Laplacian exists and satisfies the semigroup property (0.1) k(t, x, y) = ∫ M k(s, x, z)k(t− s, z, y)dvz for each 0 ≤ s ≤ t. The problem of estimating the heat kernel for small time has a long history with at least four distinct approaches. The traditional Minakshisundaram-Pleijel expansions give parametrixes pl satisfying k(t, x, y) = pl(t, x, y) +O(t ) as t → 0. In the geometric analysis literature upper bounds on k(t, x, y) valid for all t were given by Cheng, Li and Yau [CLY] [LY], with further notable work by Cheeger, Gromov and Taylor [CGT], Davies [D] and Grigor’yan [G] (see [G] for additional references). These have the general form (0.2) k(t, x, y) ≤ c(ε) t−n/2 e− d(x,y) (4+ε)t for any ε > 0. There is a separate literature that estimates the heat kernel using stochastic processes. This approach produces bounds with the “sharp” exponent −d2(x, y)/4t, and gives a lower bound for small time. In particular, in a brilliant but terse paper [M], S. Molchanov used stochastic processes to prove a version of Theorem A below (see also the exposition [A]). This paper introduces a new and relatively simple geometric analysis method that yields bounds with sharp exponents. The key estimates are obtained using the gradient flow of the energy function on path space of M . Our first result is a global, small-time asymptotic formula for the heat kernel for points x and y away from the conjugate locus. For each L > 0, let CL ⊂ M ×M be the set of (x, y) such that y is conjugate to x along some geodesic with length at most L. Given (x, y) in the complement of CL, one is led, as explained in Section 1, to the approximate heat kernel (0.3) kL(t, x, y) = (4πt) −n/2 ∑