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.

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

2007 ◽  
Vol 49 (2) ◽  
pp. 155-165 ◽  
Author(s):  
R. AYALA ◽  
L. M. FERNÁNDEZ ◽  
J. A. VILCHES
Keyword(s):  
Finite Number ◽  
Morse Function ◽  
Dimensional Case ◽  
Morse Inequalities ◽  
Discrete Morse Function

AbstractUsing the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

WITTEN LAPLACIAN ON PINNED PATH GROUP AND ITS EXPECTED SEMICLASSICAL BEHAVIOR

2003 ◽  
Vol 06 (supp01) ◽  
pp. 103-114 ◽  
Author(s):  
SHIGEKI AIDA
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Critical Points ◽  
Energy Function ◽  
Morse Function ◽  
Riemannian Metric ◽  
The Other ◽  
Compact Lie Group ◽  
Semiclassical Analysis ◽  
Witten Laplacian

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure νλ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □λ on L2(νλ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □λ as λ→∞ by a formal semiclassical analysis.

scholarly journals AN ANALYTIC APPROACH TO THE STRATIFIED MORSE INEQUALITIES FOR COMPLEX CONES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350100
Author(s):  
URSULA LUDWIG
Keyword(s):  
Morse Theory ◽  
Singular Points ◽  
Analytic Approach ◽  
Morse Inequalities ◽  
Singular Spaces ◽  
Analytic Proof ◽  
Morse Functions ◽  
Complex Cone ◽  
Witten Deformation

In a previous paper the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.

scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

scholarly journals Novikov type inequalities for differential forms with non-isolated zeros

1997 ◽  
Vol 122 (2) ◽  
pp. 357-375 ◽  
Author(s):  
MAXIM BRAVERMAN ◽  
MICHAEL FARBER
Keyword(s):  
Critical Points ◽  
Differential Forms ◽  
Explicit Estimate ◽  
Morse Inequalities ◽  
De Rham Complex ◽  
Analytic Proof ◽  
Rham Complex ◽  
De Rham ◽  
Flat Bundle ◽  
Witten Deformation

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R. Bott) and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.

scholarly journals MAPPING PROPERTIES OF HEAT KERNELS, MAXIMAL REGULARITY, AND SEMI-LINEAR PARABOLIC EQUATIONS ON NONCOMPACT MANIFOLDS

2006 ◽  
Vol 03 (04) ◽  
pp. 599-629 ◽  
Author(s):  
ANNA L. MAZZUCATO ◽  
VICTOR NISTOR
Keyword(s):  
Linear Equations ◽  
Parabolic Systems ◽  
Complete Riemannian Manifold ◽  
Holomorphic Semigroup ◽  
Bounded Geometry ◽  
Finite Speed Of Propagation ◽  
Noncompact Manifolds ◽  
Linear Parabolic Equations ◽  
Speed Of Propagation ◽  
Uniformly Bounded

Let [Formula: see text] be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form [Formula: see text]. In particular, we establish results on the distribution kernels and mapping properties of e-tLand (μ + L)s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the Ws,p-Sobolev spaces on M and E. We also prove that L satisfies maximal Lp–Lq-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂tu + Lu = F(t, x, u, ∇ u).

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