A counterexample to the Yamabe problem for complete noncompact manifolds

Lecture Notes in Mathematics - Partial Differential Equations ◽

10.1007/bfb0082927 ◽

1988 ◽

pp. 93-101 ◽

Cited By ~ 6

Author(s):

Jin Zhiren

Keyword(s):

Yamabe Problem ◽

Noncompact Manifolds

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Infinitely many solutions to the Yamabe problem on noncompact manifolds

Annales de l’institut Fourier ◽

10.5802/aif.3172 ◽

2018 ◽

Vol 68 (2) ◽

pp. 589-609 ◽

Cited By ~ 2

Author(s):

Renato Bettiol ◽

Paolo Piccione

Keyword(s):

Infinitely Many Solutions ◽

Yamabe Problem ◽

Noncompact Manifolds

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The Zero Scalar Curvature Yamabe problem on noncompact manifolds with boundary

Indiana University Mathematics Journal ◽

10.1512/iumj.2006.55.2733 ◽

2006 ◽

Vol 55 (4) ◽

pp. 1449-1460 ◽

Cited By ~ 2

Author(s):

Fernando A Schwartz

Keyword(s):

Scalar Curvature ◽

Yamabe Problem ◽

Manifolds With Boundary ◽

Noncompact Manifolds ◽

Zero Scalar Curvature

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WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000232 ◽

2021 ◽

pp. 1-38

Author(s):

Xianzhe Dai ◽

Junrong Yan

Keyword(s):

Essential Role ◽

Morse Function ◽

Noncompact Manifold ◽

Morse Inequalities ◽

Bounded Geometry ◽

Noncompact Manifolds ◽

Witten Laplacian ◽

Relative Cohomology ◽

Witten Deformation ◽

Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

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