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Nets on a Riemannian manifold and finite-dimensional approximations of the Laplacian

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Komorowski J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction......................................................................................................................................... 5 Chapter I. Families of nets on a Riemannian manifold............................................................. 8 1. Family of canonical triangulations of $R^m$...................................................................... 8 2. Non-degeneracy in the case of nets defined by simplicial subdivisions...................... 9 3. Auxiliary lemmas....................................................................................................................... 13 4. Proofs of the auxiliary lemmas............................................................................................... 14 5. Nets defined by successive simplicial and standard geodesic subdivisions............. 18 6. Non-degeneracy in the case of nets defined by standard geodesic subdivisions...... 25 Chapter II. Finite-dimensional approximation of the Laplacian................................................ 44 7. Difference forms on a net........................................................................................................ 44 8. Integration. The Stokes theorem........................................................................................... 48 9. Discrete Laplacians on a Riemannian net. The Hodge theorem................................... 52 10. Orientation and Hodge operators on a Riemannian net................................................ 54 11. Approximation of the operator d........................................................................................... 57 12. Approximation of the operator ∂ and the Laplacian......................................................... 64 13. Convergence of the approximations................................................................................... 70 References......................................................................................................................................... 79

A minorization of the first positive eigenvalue of the scalar laplacian on a compact Riemannian manifold

100%

Komorowski J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction.......................................................................................................... 5 1. A parametrix of tho laplacian................................................................................ 7 2. An estimation of the differential of an eigenfunction of the laplacian......... 16 3. A normal chart on a neighbourhood of a geodesic........................................ 27 4. Minorization of the first positive eigenvalue of the laplacian......................... 41 References................................................................................................................. 55

Ograniczanie wyników

2 Komorowski J.

1 1980

1 1979

Wyniki wyszukiwania

Nets on a Riemannian manifold and finite-dimensional approximations of the Laplacian

A minorization of the first positive eigenvalue of the scalar laplacian on a compact Riemannian manifold