Eigenvalues of the weighted Laplacian under the extended Ricci flow
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Abimbola Abolarinwa
Abstract
Let ∆φ = ∆ − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e−φdν on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.
Communicated by: F. Duzaar
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Minimal hypersurfaces in ℝn × Sm
- Higher order Dehn functions for horospheres in products of Hadamard spaces
- Unitals with many Baer secants through a fixed point
- Is a complete, reduced set necessarily of constant width?
- Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n − 1, 2)
- Classification of 8-dimensional rank two commutative semifields
- On non-Kähler degrees of complex manifolds
- Quantum Kostka and the rank one problem for 𝔰𝔩2m
- The diffeomorphism type of small hyperplane arrangements is combinatorially determined
- Superforms, tropical cohomology, and Poincaré duality
- Eigenvalues of the weighted Laplacian under the extended Ricci flow
Artikel in diesem Heft
- Frontmatter
- Minimal hypersurfaces in ℝn × Sm
- Higher order Dehn functions for horospheres in products of Hadamard spaces
- Unitals with many Baer secants through a fixed point
- Is a complete, reduced set necessarily of constant width?
- Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n − 1, 2)
- Classification of 8-dimensional rank two commutative semifields
- On non-Kähler degrees of complex manifolds
- Quantum Kostka and the rank one problem for 𝔰𝔩2m
- The diffeomorphism type of small hyperplane arrangements is combinatorially determined
- Superforms, tropical cohomology, and Poincaré duality
- Eigenvalues of the weighted Laplacian under the extended Ricci flow
