scholarly journals Infinitely many solutions to the Yamabe problem on noncompact manifolds

2018 ◽  
Vol 68 (2) ◽  
pp. 589-609 ◽  
Author(s):  
Renato Bettiol ◽  
Paolo Piccione
Keyword(s):  
Infinitely Many Solutions ◽  
Yamabe Problem ◽  
Noncompact Manifolds

Infinitely many solutions of a symmetric Dirichlet problem

1993 ◽  
Vol 20 (10) ◽  
pp. 1205-1216 ◽  
Author(s):  
Thomas Bartsch
Keyword(s):  
Dirichlet Problem ◽  
Infinitely Many Solutions

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

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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