Heat kernel upper bounds on a complete non-compact manifold
Let \(\Lambda (\nu)\) be a positive continuous decreasing function defined for \(\nu \in {\mathbf R}_ +\) and let \(M\) be a smooth connected noncompact geodesically complete Riemannian manifold of dimension \(n \geq 2\). One says that a \(\Lambda\)-isoperimetric inequality is valid for a domain \(\Omega \subset M\) if, for any subdomain \(D \subset \Omega\) the first Dirichlet eigenvalue \(\lambda_ 1 (M)\) for the Laplacian on \(M\) is bounded below by \(\Lambda (\text{Vol }D)\). The author's main result is that the existence of such a \(\Lambda\)-isoperimetric inequality for a precompact domain \(\Omega \subset M\) implies a strong upper bound for the heat kernel \(p(x,y, t)\) on \(M\), which in turn implies a lower bound for the \(k\)-th eigenvalue of the Laplacian on \(M\). The upper bound for the heat kernel is in terms of a function \(V(t)\) defined by \(t = \int_ 0^{V(t)^ -} (\nu \Lambda(\nu))^{-1} d\nu\). Much of the work involves the growth properties of the function \(V\). The author's estimate \(p(x, y, t) \leq C(V(ct))^{-1} \exp (-r^ 2/Dt)\) for \(t > 0\), positive constants \(c\), \(C\) and \(D > 4\) arbitrarily close to 4, includes many previously known estimates, including those of Varopoulos, Cheng, Li and Yau, and Davies. For further details see the paper.
scientific article; zbMATH DE number 647733