; TeX output 2002.03.04:1139 썠:-iӍ4vDt q G cmr17HeatBkuNernelonconnectedsumsofRiemannian gmanifolds#HXQ ff cmr12Alexander/Grigor'yan= !", cmsy10 6Department/ofMathematicsWW180/Queen'sGate[Huxley/Building[YImpdCerial/CollegeVtLondon/SW72BZZ_United/KingdomNDa.grigoryan@ic.ac.ukLaurent/Salo-Coste=y ĻCNRS,/T4oulouse,Franceand*Department/ofMathematics68Malott/Hall"Cornell/University,Ithaca,/NY14853-42010?United/StatesYlsc@math.cornell.edu l RMay/1999FJ' N G cmbx121D(Inutro =ductionb#' XQ cmr12This¬eisabSouttheheatkrernelonaconnectedsum"g cmmi12MH ofnon-compactman- ' ifoldsTM |{Y cmr81;M2;:::;M#2 cmmi8k x7assumingTthatoneknorwsenoughabSouttheheatkernels' foreacrhMiindividually(whichisthecasewhenMiarecompletemanifoldsof' non-negativre=RiccicurvXature).WVeannounceherematchinguniformuppSerand' lorwerbSoundsfortheheatkrernelonsuchmanifoldsM@.QTheproSofswillbegivren' elsewhere.8FVorҞanarbitraryRiemannianmanifoldM@,mdenotebrytheLaplaceopSerator' ofPtheRiemannianmetricofMandbryp(t;x;yn9)Ptheheatkernel,othatis,the' smallestpSositivrefundamentalsolutiontotheheatequationut=URuon. msbm10R+ Y%!", cmsy10=M' (hereQx;yr2Mandt>0).mAlternativrelyV,kWp(t;x;yn9)QcanbSedenedasthekernel' ofthesemigroupexp(Ut)0/.YVetanotherdenitionofp(t;x;yn9)isthatitisthe' densitryofthetransitionprobabilityoftheBrownianmotiononM generatedby' theopSerator.8InR2nP,theheatkrernelisgivenbytheclassicalformula$U_ +