LEADER 02944oam 2200505 450 001 996466549803316 005 20210703190409.0 010 $a3-030-67428-2 024 7 $a10.1007/978-3-030-67428-1 035 $a(CKB)4100000011763230 035 $a(DE-He213)978-3-030-67428-1 035 $a(MiAaPQ)EBC6478496 035 $a(PPN)253855748 035 $a(EXLCZ)994100000011763230 100 $a20210703d2021 uy 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aLiouville-Riemann-Roch theorems on Abelian coverings /$fMinh Kha, Peter Kuchment 205 $a1st ed. 2021. 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$d©2021 215 $a1 online resource (XII, 96 p. 2 illus., 1 illus. in color.) 225 1 $aLecture Notes in Mathematics ;$vVolume 2245 311 $a3-030-67427-4 320 $aIncludes bibliographical references and index. 327 $aPreliminaries -- The Main Results -- Proofs of the Main Results -- Specific Examples of Liouville-Riemann-Roch Theorems -- Auxiliary Statements and Proofs of Technical Lemmas -- Final Remarks and Conclusions. 330 $aThis book is devoted to computing the index of elliptic PDEs on non-compact Riemannian manifolds in the presence of local singularities and zeros, as well as polynomial growth at infinity. The classical Riemann?Roch theorem and its generalizations to elliptic equations on bounded domains and compact manifolds, due to Maz?ya, Plameneskii, Nadirashvilli, Gromov and Shubin, account for the contribution to the index due to a divisor of zeros and singularities. On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and Pinchover provide the index of periodic elliptic equations on abelian coverings of compact manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity. A natural question is whether one can combine the Riemann?Roch and Liouville type results. This monograph shows that this can indeed be done, however the answers are more intricate than one might initially expect. Namely, the interaction between the finite divisor and the point at infinity is non-trivial. The text is targeted towards researchers in PDEs, geometric analysis, and mathematical physics. 410 0$aLecture notes in mathematics (Springer-Verlag) ;$vVolume 2245. 606 $aDifferential equations, Elliptic 606 $aRiemann-Roch theorems 615 0$aDifferential equations, Elliptic. 615 0$aRiemann-Roch theorems. 676 $a515.353 700 $aKha$b Minh$0791289 702 $aKuchment$b Peter$f1949- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bUtOrBLW 906 $aBOOK 912 $a996466549803316 996 $aLiouville-Riemann-Roch theorems on Abelian coverings$92831540 997 $aUNISA