scholarly journals Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary

2021 ◽  
Author(s):  
Tong Wu ◽  
Jian Wang ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Compact Manifolds ◽  
Manifolds With Boundary

AbstractIn this paper, we obtain two Lichnerowicz type formulas for the Dirac–Witten operators. And we give the proof of Kastler–Kalau–Walze type theorems for the Dirac–Witten operators on 4-dimensional and 6-dimensional compact manifolds with (resp. without) boundary.

scholarly journals A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Dirac Operators ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Spectral Action ◽  
Gravitational Action

We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

scholarly journals Modified Novikov Operators and the Kastler-Kalau-Walze-Type Theorem for Manifolds with Boundary

2020 ◽  
Vol 2020 ◽  
pp. 1-28
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Spectral Action ◽  
Witten Deformation

In this paper, we give two Lichnerowicz-type formulas for modified Novikov operators. We prove Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) a boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

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