The article is concerned with the heat kernel on a connected sum \(M\) of non-compact manifolds \(M_1,M_2, \dots, M_k\) under the assumption that enough information is provided for the heat kernels for each \(M_i\) individually (this occurs for example when \(M_i\) are complete manifolds of nonnegative Ricci curvature). The authors state some matching uniform upper and lower bounds for the heat kernel on such manifolds \(M\).
The proofs are claimed by the authors to be given elsewhere.