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Books > Science & Mathematics > Mathematics > Numerical analysis
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Liouville-Riemann-Roch Theorems on Abelian Coverings (Paperback, 1st ed. 2021)
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Liouville-Riemann-Roch Theorems on Abelian Coverings (Paperback, 1st ed. 2021)
Series: Lecture Notes in Mathematics, 2245
Expected to ship within 10 - 15 working days
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This 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.
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