Evolution and Monotonicity of Geometric Constants Along the Extended Ricci Flow

Saha, Apurba; Azami, Shahroud; Hui, Shyamal Kumar

doi:10.1007/s00009-021-01848-9

Cited by 6 publications

(4 citation statements)

References 6 publications

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“…Our results can be further extended to even more complex cases by introducing the weighted p-Laplace operator (for a detailed study, see [59,60] and the references therein) to the heat equation. One can also improve the estimation by decreasing the number of restrictions.…”

Section: Future Aspectmentioning

confidence: 88%

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Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

Bhattacharyya

Azami

et al. 2023

Mathematics

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The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

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Section: Future Aspectmentioning

confidence: 88%

“…where α is a time-dependent constant called the coupling constant and ψ ≡ ψ(t) is a smooth function on M. Such a flow has also been studied in [59]. In this flow, we have…”

Section: Extended Ricci Flowmentioning

confidence: 99%

Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

Bhattacharyya

Azami

et al. 2023

Mathematics

Self Cite

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“…There are many papers on the evolution of the eigenvalue of geometric operators under geometric flows, see [3], [8], [12], [20], [22], [23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Evolution of eigenvalue of the Wentzell–Laplace operator along the conformal mean curvature flow

Azami

2024

Arkiv För Matematik

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In this paper, we investigate continuity, differentiability and monotonicity for the first nonzero eigenvalue of the Wentzell-Laplace operator along the conformal mean curvature flow on n-dimensional compact manifolds with boundary for n≥3 under a boundary condition. In especial, we show that the first nonzero eigenvalue of the Wentzell-Laplace operator is monotonic under the conformal mean curvature flow and we find some monotonic quantities dependent to the first nonzero eigenvalue along the conformal mean curvature flow.

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“…Azami, in [7], showed that the first eigenvalue of the Witten-Laplace operator is monotonic along the Ricci-Bourguignon flow with some assumptions and in [8], studied the evolution of the first eigenvalue of the weighted p-Laplacian along the Yamabe flow. Different geometric operators along different geometric flows on Riemannian manifolds are also studied in [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%